Does Romanian have any expression similar to Spanish (yo) me llamo or French je m'appelle?
yes, it is ”mă cheamă Ana” ”pe el îl cheamă Mihai”- same as all the other languages of Latin origin or influence, this comes from the Romans... This is typical Latin "missing subject" (or "understood" subject, how do you call in English "subiect subînțeles"?) sentence, in fact it is a short for the expression "(pe mine) (lumea) mă cheamă/strigă Ana", or respectively "pe el (lumea) îl strigă/cheamă Mihai", i.e. "(people) call me Ana", respectively "(people) call me Mihai"... Don't you remember the spaghetti westerns? "They call me Trinity".... hehe
We can also say "mă numesc Ana” - "(I) call (myself) Ana" / "(I) name (myself) Ana"