Topologically, the meaning of "the identity map on X is a homeomorphism" is that two seemingly different topologies are actually the same topology.

In your case, you'll have different metrics, but you'll show that the identity is a homeomorphism, meaning that despite the different ways of measuring distances, the topologies those metrics induce (via their open balls) are the same. The way you show that is to show that every open ball in one metric is open in the topology induced by the other metric (but perhaps not an open BALL anymore), and visa versa.