To register for taxation, yes.
Wasn't Joseph's family sort of well-to-do?
The Gospel is not explicit on that question, but there are hints. First, Joseph was "of the house of David," which means his extended family was hot stuff, even if he was (as it were) from the hillbilly branch.
Second, St. Luke tells us that the sacrifice they offered for their purification was "a pair of turtledoves, or two young pigeons." In Leviticus, the sacrifice prescribed is a lamb; the birds are an alternative for those "too poor to afford a lamb." So we know Mary and Joseph were short of cash, at least at that time (40 days after Jesus' birth.)
Finally, in Luke 2, we learn that "His parents went to Jerusalem every year at the Feast of the Passover." Obviously, this required some financial resources, and probably well-off relatives in Jerusalem with whom they would board during the feast. The overall picture is of reasonably well off working people, with fancy relatives.
The sacrifice of turtle doves always seemed incongruous for a family that otherwise seems fairly well-to-do. Perhaps they were cleaned out by the taxes they had to pay in Bethleham.