Can an user log into any phone, and get their extension number on that phone?
I found that Druid has this capability (they call it “Virtual Extensions”), and I want to know if it can be done in Asterisk without Druid.
Yes Asterisk can do it. Theres some other things you need to think about such as the original extension that is assigned to that phone. What do you want those calls to do for the original extension?