I have a brother who always sing in the bathroom, singing in the bath, or singing in the shower, is a widespread phenomenon.
I cannot carry a tune so I do not do that.  I wonder why people do that.
Its fun though, because Many people sing in the bathroom because the hard wall surfaces, often tiles or wooden panels, create an aurally pleasing environment.
Do people turn on the radio in the shower and sing to the songs played?
Oh yes, some use phones, The multiple reflections from walls enrich the sound of their voices
Do people sound better to themselves than they sound to other people outside the shower?
Yes, the bathroom has an echo, though This habit was reported centuries ago.