GFCI tester is tripping 15A breaker in panel; outlets on circuit do not appear to be GFCI outlets - is this a problem?
Worth noting(?) this 15A breaker that keeps tripping is the only one in the panel with a test button.

Because it is a GFCI breaker.
Yup.  That's a thing.
The GFCI rabbit hole gets weirder: Any GFCI device (recep, breaker, deadfront, switch, combo switch/recep) can protect downline circuits.  So if you have a GFCI recep at spot 1, and cable going onward to spot 2, there is an option to wire it so the GFCI recep at 1 protects both the cable to 2 and the outlet at 2.  Or 3 or beyond.
In fact that is exactly what a GFCI breaker is doing: protecting the whole circuit.
The reason this wasn't apparent to you is that the receps did not have the mandatory sticker: GFCI Protected.  As such, they were a code violation and need to have the sticker added.
This puts the &quot;random trips&quot; in a whole new light.
It's simply ground-faulting of the various appliances.  Keep in mind that small leakage from several appliances can add up to enough leakage to trip a GFCI device.  So an additional appliance (like the vacuum) can &quot;push it over the edge&quot; as it were.
However, a 2-prong appliance is unlikely to cause a GFCI trip unless it is ground faulting internally and it contacts something grounded.  (E.G. if the vacuum tripped everytime you vac'd under a water pipe or something).
If you have a cyclical load like an air conditioner that sometimes trips it, then the trip will appear to sporadically occur with any appliance you regularly plug in: the appliance isn't at fault, the other load just happened to choose that moment to trip it.
If unrelated circuits trip it, it's possible the 2 circuits' neutral wires have been bridged together at some point.  Current from that circuit returning on this circuit's neutral will cause a trip.
Lastly, it's not impossible for a GFCI to fail outright.  I strongly discourage people from thinking in those terms, because experience has shown most people will instantly leap to that conclusion: a version of &quot;shoot the messenger&quot;. I am glad you are not vulnerable to that.
It wouldn't hurt to buy a second GFCI breaker with an aim to fit it on a second circuit, and use it to test this circuit before assigning it that task.