The behaviour you're seeing is the synonym system itself being broken.
For implementation reasons, all of the behaviour you've seen, and the behaviour seen by others in the comments, is caused by tag synonyms both existing and not existing at the same time.
You would think a synonym means "all of these tags are now tagged that instead", but it doesn't. internally, yes, but this isn't reflected until an edit is attempted. This means that, even though the tags are synonyms, they aren't fully the same tag. This is why search is broken, and this is why you don't currently have a tag score of 1500+ in selenium-webdriver; the score from selenium didn't transfer, so you didn't get that tag badge.
Even if it fully transferred, you risk having the inverse problem; being unable to close selenium questions with selenium-webdriver. There are ways to fix this in the system, but that requires dev time.
Historically, the solution has been to merge tags. Tag mergers involve doing what you think a synonym did, but it also does so irreversibly. This is why many tags don't get merged; if it turns out the synonym was bad later, it's easier to remove a synonym than it is to unmerge. Unmerging tags isn't possible without dev intervention.
Merging involves a database-level rewrite, which in this case would force all selenium questions to instead be tagged selenium-webdriver. I attempted to do this for selenium, but computer said no. The query (presumably) timed out with an HTTP 500 and a generic "an unknown error occurred". Because it's a database-level rewrite, there's a lot of stuff that gets rewritten. In addition to the 100599 questions (0.4% of all questions!) needing to be rewritten, there's also all the revisions associated with those questions, as well as some other things I don't remember. There's a lot of data to rewrite with that many questions, so it isn't surprising that the merger failed.
Mergers were likely introduced as a band-aid to avoid sorting out the issues with the synonym system. This must've also been back when the site was still tiny and merging didn't involve rewriting over 100000 questions and probably at least a few hundred thousand revisions.
It's therefore possible that the behaviour observed here is by design; that the intent was for mergers to be done early and often. A merger would fix all the problems presented here, and it wouldn't require any dev time (... disregarding the dev time required to perform the merge, seeing as the mod tool likely times out too fast for it to work). So to answer your question:
On Stack Overflow, I possess the selenium Gold badge, but the permissions don't seem to apply to its synonyms. Does this selenium badge also apply to its synonym selenium-webdriver? Or do I need to unlock a badge for that tag?
Under the current system, yes. You shouldn't need to, because the score should transfer, but a full transfer would mean you risk not being able to close selenium questions anymore, so we're still back to tag badges needing to respect synonyms.
Simply put, this is a consequence of tag synonyms being poorly handled in every single part of the system, and it's absolutely a bug, but so is the rest of the synonym system.
