Focusing on the Electron and Unity situation, in general sharing tokens issued to one application in another is not correct. However, if this is a case that in fact both Electron and Unity is perceived to the end-user as the same application and the different frameworks are just an implementation detail than it could be an exception to the above guideline.
As you mentioned, given that the access tokens being issued are bearer tokens from a technical perspective any party that posses the token would be able to call the associated resource server with success; this means that the 403 might be some implementation issue. In conclusion, depending on the exact characteristics having Unity act as an independent client application that takes the user through login again (even if SSO can be leveraged) could the best approach, but in some scenarios the reuse of tokens could be feasible. What I would likely never do would be enter the realm where you’re issuing tokens yourself in your own systems.