There is video evidence of me playing Street Fighter 2 back when I was like 3. I have no actual memory of doing so--my parents hid it away because they thought it'd make me violent.
I'd sort of get back into FTG with Smash Bros. + Melee, but that doesn't count.
So really, my first fighting game is actually like... really late with Guilty Gear XX #Reload (Xbox) and Melty Blood Re-Act Final Tuned shortly after, probably.