Why didn’t Marc Anthony have a traditional Roman name?

He was a member of gens Antonia, his father's name was Marcus Antonius Creticus and his grandfather's Marcus Antonius "Orator". Marcus was a common Roman name. Actually, along with Marc Antony and Augustus, the third member of the second triumvirate was Marcus Aemilius Lepidus.

Edit: it isn't rare that names of historical figures change in different languages (e.g. the English philosopher Francis Bacon in Italian is called "Francesco Bacone")

And Thomas Moore -> Tommaso Moro.

But this happens to Latin names as well, Marcus Antonius is Marco Antonio in Italian and Mark Antony in English.

In Italian there is also the (derived) "marcantonio" with the meaning of "big guy".

he did. his name was Marcus Antonius.

we know him as Mark Anthony because of Shakespeare.

His anglicized name is spelled Antony (without an ‚h‘.

What's wrong with Marcus Antonius?