Odo and Laas were given names by the solids that they'd encountered.
LAAS: The Varalans called me Laas. In their language it means changeable. Not very imaginative, is it?
ODO: At least it's appropriate. My name means unknown sample. The scientist who found me didn't know what I was.
When he finally meets other changelings, he learns that they don't normally communicate using spoken words, preferring a direct mingling of minds in a small version of the Great Link or simply within the Link itself.
The short answer appears to be that the changelings don't typically use names other than generically identifying themselves as a Founder to their acolytes.
The Star Trek RPG Module Book 5 (Aliens) had this to say on the subject;
They have no need of their own language, for their merging can
provide far greater and more immediate communication than words could
possibly convey. Changelings adopt names for themselves when operating
among "solids," but have little use for them on their homeworld.