When something is described as assumed, it's fake. An assumed name is made up, often to hide a true identity.
If you're living under an assumed name, you've taken on a new, fictitious name to disguise yourself or simply to start over as a new person — to give yourself an assumed identity. An assumed facial expression is false, and an assumed address isn't your real one. The adjective assumed comes from the meaning of assume that's "take on or adopt, often falsely," from the Latin assumere, "to take up."