2-step verification with Gmail only applies when you're accessing it in a browser. If you're using a mail app, once you sign in, you should be good.
I did run into the 2-step thing today when I was fixing a problem on my son's MBP and had to log into my own Gmail account in a browser to get a password. So yes, they have indeed implemented it. I expected the verification to go to Messages like everything else does, but nope - it went to the Google app on my phone (which I never use and had to search for!) - it basically requires you to tap "ok" and that's it. No numbers to type in or anything like the bank/CRA etc. So better in that regard, though does mean you need to have the Google app installed.