Is mig welding a forge welding?

No, Forge welding is the simplest form of welding where two metals are heated to a state than can be forced to meld into each-other such as hammering or by other mechanical means such as a press.

MIG Welding is short for Gas Metal Arc Welding (GMAW). This is a process which uses electricity to melt the parent metal and the electrode (wire) while shielded by a inert gas to keep contamination's (our atmosphere (aka air)) out of the weld puddle.

Forge welding uses brute force and heat to join metals while MIG welding uses electricity and a shielding gas.