I think the appropriate question is "is it likely", rather than "is it possible". A lack of matches to a given family name is suggestive, but it's also possible that nobody from that family has submitted a sample (I say possible, it's hard to tell how likely without knowing, say, how large the pool of living descendants is).
There other possible (although admittedly less likely) scenarios that could account for the DNA results, and, well, maybe his parents changed their mind about his given name after the birth certificate was filed, or maybe he himself just didn't like his given name.
Determining whether/how you're related to a particular set of ancestors is a job for the DNAPainter "What are the odds" (WATO) trees:
DNA Painter WATO tool
Although I've never used it this way, if you build a tree starting from one of your grandfather's ancestors of record, and include any known descendant samples (you need to know they exist, and that you don't match them), you should find out fairly quickly if there's zero chance they're your biological ancestors.
Similarly, if you can find a fairly close common ancestor among the samples you do match, you should be able to build a WATO tree of those matches, and get a fairly good sense of where your grandfather might fit. You might find more than one such ancestor, and build more than one corresponding WATO tree (bio great-grandfather's family vs. bio great-grandmother's, or further back than that).
So, proving the biological ancestors and proving adoption aren't quite the same thing, although the former would certainly make the latter a more likely possibility.
