No, they do not mean exactly the same thing. Dropping "off" doesn't make it incorrect but there is a nuance of meaning.
fight something/someone off - would mean "succeed in overcoming (so as to be removed);"
fight something/someone - struggle to overcome;
More to the point, "fight off" carries a slightly different meaning where "off" changes the meaning as in:
finish off - completely defeat something or someone; completely consume food or drink;
finish - to complete something; finish work, homework or something. So if you said
"I finished off my homework" that would be figurative (though finish with is more common), yet it's not
something I'd see commonly used.
Sometimes different verbs match different nouns.