That's probably because your work account is a Google Apps account, which works much better especially for Drive (the problem is that you can't have more than one non-Apps account using drive at once in the same session. I have no idea why, but it seems to be deliberate). I'm unsure if the brokenness is fixed with Apps accounts on other Google services that are bad at handling multiple accounts.
Support for multiple accounts in drive is pretty great (minor papercuts here and there) as long as all but one of your accounts is an Apps account. This feels like an issue that they've fixed but not rolled out.
Except this has been a problem for years, and according to a friend of mine who's had a "personal" Apps account from back when it was free, never has been a problem for Apps users, so I'm not sure what's going on on their end.
stackdriver.com has to be one of the worst! selecting the right account just causes an infinite loop if the correct account isn't index[0] in your logged in accounts!
ADDED: In general a lot of services don't handle multiple accounts terribly well. I've had issues with Wordpress as well. I actually have two different soft-key authenticator programs related to making personal and work accounts play nice.
Sounds like you're using the Google Accounts feature, and using the [separate] browser profiles feature would exactly solve your problem.
In particular: you can log into different profiles in the browser (each window can be associated with up to one profile, and they can be different), by clicking the little "Sign In" tab in the upper right corner. See also https://support.google.com/chrome/answer/2364824?hl=en&co=GE....
You can then open tabs under the different profiles with right-click, "Open link as", and selecting the appropriate profile.
(Personally, once I tried profiles, I never touched multiple-accounts-logged-in again.)