Since you asked how it felt to be on the other side, I thought I'd give you the point of view of someone who's been in this position.
I'm support staff for a bunch of scientists at a major state university. The person who did my job before me had a similar name to me. I have never met her (she was gone for a year before I started) but one of the scientists has pretty consistently called me "Chris" (her name was Christine). Not always, I think. He knows that it's not my name, so it's not quite the same situation... perhaps it is worse. I'm probably 10 years his junior and, while I don't work for him directly, I see him on a daily basis.
In general, he does it on his way home. "Have a good weekend, Chris". Most of the time he doesn't address me by name.
At first I assumed that I was misunderstanding him. That lasted the first couple of times. Eventually... after several months... it became clear that he probably just hadn't assimilated that he was saying the wrong name.
Today I corrected him. He was standing and talking near my desk with our project manager and I was occasionally joining in the conversation. They both started heading out. He said "Have a good weekend, Chris" and I said, simply and without any sort of inflection, "Catherine". He stopped, did a double take and finally realized what he'd done... and probably had been doing for a while... and apologized.
And that was it.
The fact that this PA didn't correct you (to me) says something but it doesn't mean you can use it to understand how he felt. All you can really know is that, in your interactions, he wasn't interested in correcting you but you can't know whether it is because he doesn't care or simply didn't want to make you uncomfortable.
I am pretty non-confrontational. I would rather sidestep an argument than cause one... and this isn't even the sort of thing that would cause an argument... If I had been calling someone the wrong name for months I would have been mortified, so confronting him, to me, meant that I'd embarrass him. I was happier, for a time, to assume that I was hearing wrong than that he was calling me someone else's name.
Not everyone is like me. I'm very particular about my name. This is generally due to the fact I have a name that gets spelled about 50 different ways and it's rare that anyone gets it "right". So, even despite this, I didn't say anything. Some people don't care or they just assume it's better to leave a good impression than to speak up and cause them to lose face somehow.
So, to answer your question
This question is really: would correcting me have seemed like a big deal to him, or was this mistake of mine so minor that he didn't think it worth mentioning?
There's no way to know. Probably a little bit of both. If he's trying to make a good impression because you might hire him full time or he can use you as a reference, it may be better for him to let his boss do what she did. Correct you in a passive way by asking "How did Mitchell do?" It's completely possible that he texted your lawyer after he left and said "It went great but he called me Michael the entire time".
DC is a scary place with some very influential people. I lived there for a year before I decided it wasn't for me. You never know who you're going to be working with when you're there or how they're going to react - or overreact to things.
What can you do in the future? Get their name in writing. If you can't pronounce it, ask. If you see Mitchell again, tell him that you misheard his name and encourage him to speak up in the future.
In closing, all I can say is this:
Call me anything you want as long as it's not "Late for dinner".
