Ohhhh no. Mitsuki can feel the whole atmosphere change after his question. He didn't want to do that but even before Akihiro speaks, he understands that he crossed some kind of line. Considering how important boundaries are in his own profession, he feels a little like he should've anticipated that, but Akihiro doesn't seem upset. He also doesn't back off, and Mitsuki's almost embarrassed at how relieved he is by that.

The name thing just has him making a face--clearly the client wasn't Japanese--but it turns into a proper frown when Akihiro starts talking about whoever comes after or Mitsuki wanting him gone. "What if that's not what the client wants, though? Sure, you work for your own company and my management more than for me, but they're not the ones here right now. You might've saved my life! You deserve a chance to work through that just as much as I do, or at least to unwind with a drink and somebody to talk to." Sure, he's probably projecting a little trying to imagine how he'd deal with laying low like this if he couldn't be casual with Akihiro, but he feels pretty strongly about it regardless. They're safe here. Why can't they both be off the clock?

Actually... "What if I said you were fired until tomorrow morning? Then the rules wouldn't apply."