Firstly there's a reason why they are referred to as Ethnicity Estimates. This is a consistent refrain from several commentators. There are plenty of instances of siblings (who must by definition have the same ancestors) who have different ethnicity estimates.
Secondly the estimates depend on who has submitted their DNA to Ancestry etc. This means that there is only a small reference population for France because essentially they ban consumer DNA testing, so someone with genuine French ancestry may not match France. I am unclear how much DNA testing has been done in Germany - I had an idea that it wasn't prevalent in Germany so unless someone matches a small number of Germans, they may not be ethnic Germans in Ancestry etc. Comments gratefully accepted.
I can speak from odd experience here. I'm 1/8 Scottish and 1/16 Irish and my Ancestry ethnicity gave a small percentage of Scandinavian ancestry initially. Then the same percentage went to Germany. Now it's gone - where? Dunno but my Irish percentage is suddenly greater than my Scottish. It doesn't fill me with confidence in German or Scandinavian ethnicity estimates.
My rule of thumb is simple - if the paper conflicts with the ethnicity estimates, believe the paper.
To add a couple of points to the above...
"The Legal Genealogist" (Judy G Russell) has blogged several times on ethnicity estimates. For instance, see And Still Not Soup... from 2019. What is particularly relevant to the original query is her statement that:
My own results are — literally and figuratively — all over the map. I’m German with some companies, not German at all with another. Largely Scandinavian with one, only slightly Scandinavian with the others.
In other words, it might be that the same holds for the OP's husband and what one company calls Scandinavian, another might call German. Also perhaps of interest is AncestryDNA ethnicity estimates updated, from 2018, which mentions a prior update to its ethnicity estimates by Ancestry, which moved her from 31% Scandinavian to 19% Germanic Europe, 7% Norwegian and 6% Swedish. (She has German ancestry, by the way).
Point 2 was me wondering whether there might be a substantial amount of Scandinavian ancestry in the peoples of North Germany anyway - it's only the width of the Baltic away!
In order to take this forward, I would suggest that ethnicity estimates be put to one side for the moment, and the OP concentrates on looking for "cousin-matches" for her husband. If matches appear that seem to be related to the husband's maternal GM, then the obvious conclusion is that he is indeed related to his GM. (NB - "obvious" is not the same as "correct" if the link is via another set of relationships).
A final point is in relation to "We have no birth certificate for her". According to the FamilySearch Wiki for Michigan VRs
County registration of births in Michigan began in 1867 and was generally complied with by 1915
I am always concerned when it says "generally complied with...". I have no knowledge of percentage coverage for Menominee birth certificates but would suggest that the OP needs to investigate that aspect if discovery of a certificate were felt to add anything.
