Update: Here it is, my precious:
Just invite your usual Apple ID in each desired team’s Users and Roles section, click a link from the invitation email and you’re done! 🎉
It really seems that you can’t use the same iTunes Connect account for multiple teams.
But what I’ve recently found out is that you can have multiple iTC accounts under the same email by using plus-tagged emails for registration. Thus, my iTunes Connect account IDs look like
myusername+someteam@gmail.com and myusername+anotherteam@gmail.com, which means two different email addresses for Apple but actually a single email account for me.
Would be happy to hear about any possible downsides of such an approach, though I really find it more convenient than dealing with a bunch of emails and Apple IDs for different dev teams.