Siri's language choice (Settings) - does it affect speech recognition?
In my experience with Siri, the initial language setting of Siri does help a lot to help it discover your accent more quickly, but it's not the only measure it uses. It will base its understanding of your accent off of its own starting point. It's pretty difficult to convince Siri you're not American if you choose American English as its language. It will eventually change, but it takes a lot of time, and might not even be completely accurate.

I speak with a pretty standard British accent. Not northern, not southern, almost Welsh, actually. Siri's accuracy shot up with I switched to a British starting point.

I suspect that over time, Siri will eventually categorise your accent correctly regardless of its own language setting. This is evidenced by the fact that after a few tries, some of contacts' names (ones that a computer would have a hard time pronouncing correctly) are recognised now whereas they weren't originally.

The key is that Siri will attempt to categorise your voice into an accent. It will not necessarily learn your own specific accent, but it will broadly put your accent into a box, with a few exceptions for how you pronounce proper nouns like places, sports teams, people, and even apps.