The only other Changeling who’s not disguised as something or someone else is the Female Changeling (Salome Jens) who says she chose her appearance to make Odo feel more comfortable. That doesn’t explain why she doesn’t use another face when Odo isn’t around, but honestly that’s probably just visual shorthand to remind us who she is.The one thing I wish they'd explained is why all the Changelings look like Odo, when Odo is attempting to look like the scientist who raised him. I wish they'd thrown in at least a brief explanation of "we took on this form to try and convince you to stay" or something.
That said, it would certainly be useful for Changelings to have a clear and consistent appearance that does not look like Vorta or Jem’Hadar, to be obviously Founders, to avoid confusion, and to consistently trigger their obedience conditioning. And maybe that’s what Odo’s look is based on, from his own infant memories.