Yeah, shouldn't it be 'fast reciprocal square root'?

Agreed. Inverse is not a synonym for reciprocal, but it's too late, the name has stuck.

Here, "inverse" is an abbreviation of "multiplicative inverse" (aka. the reciprocal). Granted, shortening it to just "inverse" is misleading, but the name has stuck.

No, it's completely different than simple squaring. The inverse square root is undefined for negative numbers ;)

That's not right. The so-called fast inverse square root function computes the reciprocal of the square root, not the square.

You're not the only one to be confused by the name. An awful choice of name, in my opinion.