No it isn’t. He compares himself to Adam once (“I ought to be thy Adam, but I am rather the fallen angel, whom thou drivest from joy for no misdeed”) but he never calls himself that. And frankly, considering how much Frankenstein and the monster hated each other I don’t think either of them would want to share a name.
It’s a silly nitpick anyway. The monster, Adam, calls the doctor, Victor Frankenstein, his father. Surnames are inherited, thus they are both Frankensteins.
Slavoj Žižek’s Freudian-Hegelian interpretation of Mary Shelley’s story is worth investigating especially in relation to Shelley’s family, the French Revolution, and transgressive sexual politics.