Yes, there were birthday cakes in the ancient Rome.
From Wikipedia article on History of cake,
During the Roman period, the name for cake became “placenta” which was derived from the Greek term. Placenta were baked on a pastry base or inside a pastry case.
The Greeks invented beer as a leavener, frying fritters in olive oil, and cheesecakes using goat's milk. In ancient Rome, basic bread dough was sometimes enriched with butter, eggs, and honey, which produced a sweet and cake-like baked good. **Latin poet Ovid refers his and his brother's birthday party and cake in his first book of exile, Tristia.
Ovid is a Latin Poet from ancient Rome who is believed to live between 43 BC to 17 AD. He has referred to his and his brother's birthday cakes in one of his poems Tristia.
From his work Trisitia hosted by archive.org
11 What hast thou to do with Pontus ? Is it that Caesar's wrath
sent thee too to the remotest land of the world of cold ?
Thou awaitest, I suppose, thine honour in its wonted guise :
a white robe hanging from my shoulders,
a smoking altar garlanded with chaplets,
the grains of incense snapping in the holy fire,
and myself offering the cakes that mark my birthday
and framing kindly petitions with pious
TRISTIA III, XIII. A BIRTHDAY AT TOMIS
So, it is known that there were birthday cakes made from different ingredients in the ancient Rome.