You can map 1000 one dollar bills to every single 100 dollar bill and never run out, even in the limit, and therefore conclude (equally incorrectly) that the infinite $1 bills are worth more, because infinity isn’t a number.
Yes, the mapping that you’re describing isn’t useful; but that doesn’t invalidate the simple test that we can do to prove that two infinite sets are the same. And the test is pretty intuitive in some cases. If for every fork there is a spoon, then they must be an equal number. And if for every positive whole number (e.g “2”) there is a negative whole number (e.g. “-2”), then the set of positive whole numbers and the set of negative whole numbers must be the same.
Another thing that seems unintuitive but might make the concept in general make more sense is that you cannot add or do any other arithmetic on infinity.
I agree with you that adding infinity to itself just gets you the same number. But now we’re starting to ask: just what is infinity? We could think of infinity as a collection of things or we could think of infinity like we do numbers (yes, I know infinity is not technically among the real numbers). In the latter sense we could have a negative infinity {-1…-2…-3…etc.} that is quantitatively “worth less” than a countable positive infinity. On the other hand, if we think of infinity as a collection of stuff, then you couldn’t have a negative infinity because you can’t have less than zero members in the set.
I’ll admit that I’m getting out of my depth here since I know this stuff from philosophical study rather than maths proper. As with anything, I’m happy to be proven wrong, but I’m quite sure about the 1-1 correspondence bit.