We all had our own GitHub Enterprise accounts. But only one was logged on to each computer. Rather than switch out the accounts when someone else used a computer, we just included the initials of whoever was making the commit in the message. E.g. DU: Fix some bug, if Double Unplussed was making the commit.
This only really worked because no one was using a GitHub login that they also used for personal projects - only the corporate ones were connected. And it made the GitHub statistics on who was contributing what completely useless. But it did mean that we could rotate among the shared computers and push easily, and if you did need to ask someone for help when investigating a bug, you could still see who made the changes via git blame - just by looking at the message instead of who Git thought had pushed it.
Might not work for your use case, and I'm sure some readers will be horrified, but it worked well enough for us, in an XKCD 1172 (https://xkcd.com/1172/) style manner.