I can’t speak for the Greeks, but the Romans had an equivalent of “singing in the shower.” Allowing for poetic exaggeration, there are anecdotes from Roman writers that people, men I assume given the segregation of the public baths, liked to hear themselves sing in the baths. (I’m allowing here that the difference between singing or reciting poetry was a thin one.) Let me quote at length Seneca’s description of the sounds of a Roman bath (Moral Letters, 56) from a translation by R.M. Gummere (1917—sorry for the date; I’m grabbing an out-of-copyright text) that I’ve adapted here and there:

I’m done for if silence seems the most necessary thing for a man who secludes himself to study. . . . I live over a bath house! So picture all the types of sounds that could make you hate your ears. When musclemen are lifting lead weights while working out—or pretending to work out—I hear them grunting; and whenever they blow out their held-in breath, I hear them panting with wheezes and the loudest groans. If there is some sluggard content with a rubdown, and a cheap one at that, I hear the slap of the pummeling hand on his shoulder, changing sound depending on whether it’s flat or cupped. And if some ballplayer comes along shouting out the score, that is the finishing touch. Add to this the arrest of some brawler or thief and the man who likes the sound of his own voice in the bath; now add those who leap into the pool with a huge splash of water. Besides all those whose voices, if nothing else, are at normal level, imagine the depilator advertising his services with his penetrating, shrill voice, continually giving it vent and never holding his tongue except when he is plucking armpits and making his client yell instead. Then there is the drink seller with his varied cries, the sausageman, the pastryman, and all the vendors for the food shops hawking their wares, each with his own distinctive cry.

Petronius’ hilarious Satyricon gives another vivid description of singing in the baths, in this case the private baths of the nouveau riche Trimalchio, who “sat down [in the baths] as though tired, and being tempted by the acoustics of the bath (sono balnei), with his drunken mouth gaping at the ceiling, he began murdering some song by Menecretes—or so we were told by those who understood his words” (sec. 73; J.P. Sullivan’s translation). Later, another character, the poet Eumolpus, says, “You know I was almost beaten up even while I was taking my bath, just because I tried to recite a poem to the people sitting round my tub” (sec. 92).

The poet Horace also remarks on the sounds of the bath (Satire I. 4, 76): “Some people recite their writings in the middle of the forum, and also many recite while bathing--the enclosed space sweetly echoes their voice.” Martial remarks (Epigram 44) on the sounds of poetry in the baths when he accuses Ligurinus of pestering him with his poems (James Michie trans): “You read to me to me when I’m standing and when I’m sitting, / When I’m running and when I’m shitting. / If I head for the warm baths you make my ears buzz with your din, / If I want a cold dip you stop me from getting in.”

Garrett G. Fagan’s Bathing in Public in the Roman World (1999) has more to say on the subject of singing in the baths.