We probably can't prove that "Soldier Matthew" is the same as "Family Matthew", but we can seek to demonstrate that they are not the same person. If, for example, the military records show that Soldier Matthew was serving overseas at the time that Family Matthew got married, we would conclude that they were different men.
The discharge record has a hand-drawn table in the left margin, which appears to enumerate Matthew Stretch's service record:
85th Regiment 2 Years
99th Regiment 3 1/2 Years
East Norfolk Regiment 3 Years
Royal East London Militia 8 1/2 years
(The lower line is hard to read, and continues into the fold, but it reads "Royal East" by comparison with the line above, and thus is presumed to be the Royal East London Regiment, where Matthew finally served. There appears to be a line drawn under the "8 1/2" suggesting a summation.)
This gives us a total service of about 17 years (give or take a bit, obviously) ending in 1805. On the second page of the discharge record it states 18 1/2 years service instead. That can't be accurately derived from the numbers in the table, but perhaps the table figures are rounded down to the nearest half year, while their sum without such rounding is 18.5. I'll use 17 years below as it's less constrictive, and I don’t think it changes the analysis significantly.
So the very latest Matthew can have enlisted is ~1788 if all periods of service were contiguous, but I will not assume that they were (he would have been nearly 30 at such time, which is older than a typical recruit). I will, however, assume that all individual periods of service were contiguous, as otherwise there are just too many unknowns. My understanding is that "popping in and out of one's regiment" was not common practice, in any case. I will also assume that the periods of service took place in the tabulated order.
Typical enlistment ages appear to be late teens and early twenties, but younger is not unknown (even in Matthew's likely extended family). Assuming that Matthew didn't lie significantly about his age, a date of ~1775 seems like a reasonable lower bound for his enlistment date. That leaves roughly a 13-year window in which his service might have started.
So can we work out when Matthew might have served in each regiment, and does this help? This basically becomes a bit like one of those logic problems where we must (for some reason) work out which vicar went to the opera wearing a purple hat…
The 85th Regiment (2 Years)
Soldier Matthew's first regiment, the 85th Regiment of Foot, has had three incarnations:
1) 1759 - 1763 as the "Royal Volontiers"
2) 1779 - 1783 as the "Westminster Volunteers"
3) 1793 - 1881 as the "Bucks Volunteers", later the "King's Light Infantry"
Clearly, #1 is too early for Matthew, and #3 is too late, as 1793 is well past the 1788 cutoff noted above. So we can conclude that Matthew first served in the Westminster Volunteers for about 2 years in that 1779-1783 period. (Some sources suggest 1778 instead of 1779 for when the 85th was raised. This may be from ambiguity in records, or might reflect when the process started and finished. I'm not sure what the convention is here.)
Significantly, a starting date of ~1788 (from fully contiguous service periods) simply isn't compatible with joining the 85th as his first regiment, as it didn't exist then. So either Matthew's periods of service were not contiguous, or the order in the table is incorrect. As noted above, I've assumed the former and not the latter.
The Westminster Volunteers were not a fortunate regiment. They were sent to Jamaica in 1780, where:
Decimated by disease in Jamaica, what had been described as “one of the finest corps that ever left the shores of England” was reduced to 71 men in two years; sent back to England in 1782
Worse was to come:
1782 - greater part of regiment perished on board the Ville de Paris three-decker and other French prizes taken in Rodney's action with the Comte de Grasse, which were swept away by a cyclone off Newfoundland, when homeward bound
It appears that the main troop carrier sank, and few of the 85th made it home. Matthew might have been one of those lucky, battered few, or he could perhaps have transferred out before they sailed in 1830.
Summary: The likely service period in the 85th was ~1779-1780.
The 99th Regiment (3.5 Years)
The 99th Regiment of Foot also had multiple incarnations:
1) 1760 - 1763
2) 1779 - 1783 (the Jamaica Regiment)
3) 1793 - 1797 (disbanded in Demerara, men drafted into 39th Dorsetshire Regt of Foot)
4) Three more, too late to consider.
Again, #1 is too early, and #3 is too late for Matthew's service to fit, so he must have served in #2. 3.5 years service is approximately the entire duration of that regiment's existence, so he must have joined close to the start. He cannot have been in the 85th at the same time, so he must likely have moved from the 85th in ~1780 - before that regiment sailed to Jamaica, thereby avoiding their tribulations. The 99th took him to Jamaica anyway.
Summary: The likely service period in the 99th was ~1780-1783.
The 9th Regiment (3 Years)
The East Norfolk Regiment was raised back in 1685, became the 9th Regiment of Foot in 1747 and then the 9th East Norfolk Regiment in 1782. At that point the regiment was based in Norwich to build up strength, so there's a good chance that Matthew enlisted there after returning home following the disbanding of the 99th. Before that date, the 9th had been in the Americas for over a decade (including three years of captivity) until 1781, so Matthew can't realistically have been in the 9th before 1782.
A book, "The Ninth, or the East Norfolk, Regiment of Foot", compiled by Richard Cannon and published in 1848, has a timeline for the regiment which includes this relevant section (from archive.org):
1783 Designated "The Ninth, or East Norfolk" Regiment
1784 Marched to Scotland
1785 Embarked for Ireland
1788 Embarked for West Indies
1796 Re-embarked for England
If Matthew had joined the 9th shortly prior to 1788, he would have spent nearly another decade in the West Indies, which doesn't fit with his 3 years service. So he must have left before 1788. His three years fits flexibly between 1783 and 1788, and so he probably went to Scotland and/or Ireland during his service.
Summary: The likely service period in the 9th was ~1783/4/5-1786/7/8.
The Royal East London Militia
Matthew had 8.5 years of service in the Royal East London Militia in March 1805. Assuming that was contiguous, and that it ended in 1805, that means he enlisted in around 1796. He can't have joined before then without a break in service, which I have assumed did not occur.
Summary: The likely service period in the RELM was ~1796-1805
So that leaves a gap between ~1785 and ~1796. What could he have been doing in this time?
Well, if we go back to the "Family Matthew" Stretch who married Elizabeth Flight, there's a happy coincidence. Matthew and Elizabeth married in 1788 and had children from 1789 to 1804. The marriage and first few children fit perfectly within that 1785-1796 gap in military service. The Royal East London Militia would not have travelled far (an archived regiments.org page suggests <12 miles), so the continuation of their family during that period is plausible.
Now we can, tentatively, build up a parallel timeline for the two Matthews:
There does not appear to be any clear conflict between the two timelines, so we can rule out neither the "One Matthew" nor the "Two Matthews" models. Typical. Baptism records for Family Matthew's children mention an occupation of "Porter" or "Painter" (and one I can't read) but none, alas, note him as a militiaman. "Painter" is one point of overlap between the records for the two Matthews (the other being approximate age). Although that was surely not an uncommon occupation, in combination with an uncommon name it looks more relevant.
So the military record does not show that Matthew the soldier was a different person to Matthew the father. Instead it provides a rather convenient fit to the family records. This isn't proof, of course, but it starts to look reasonable given the paucity of alternatives. I still don't have a birth/baptism record for Matthew - the only good match was born in the wrong Norwich parish and died in infancy. So there may be missing records or still some errors in the above work.
There are still questions, though, that might undermine all of this. Is a ~10 year gap in service between the 9th and the Militia plausible as described above? Would there not have been a pension or other record from his earlier service period to ~1785, or would that have been replaced by the 1805 one? Are the changes of regiment reasonable? Anything that makes the service record analysis "not work" (for example, requiring a single contiguous block of service) would instantly nullify Soldier Matthew as a match for Family Matthew. That would be disappointing, but still a valuable and important result.
Even so, I don't think the service pattern as tabulated can work in a single block to 1805, as the 85th and 99th didn't exist at the right times. And even with reordering (which I haven't analysed in proper detail yet), the 9th spent 1776-1781 in the Americas, which makes it hard to fit into an alternative order.
In summary, a comparison of military and civilian records may identify conflicts that allow the elimination of candidates. This particular case yields a convenient lack of conflict, allowing the "One Matthew" model to survive alongside the "Two Matthews" model. I think the "One Matthew" tree is probably correct, but I'm keen to know if there's anything else that can be pulled out of these (or other) records, which might settle the question once and for all.