TL;DR you can't use infinity like that and your calculus professor will yell at you if you try.
Infinity isn't a real number and it's not generally useful to think of it as one like the dude in this comic is trying to. However, in calculus you can treat it as a concept that a variable or expression can approach. In that way, "approaching infinity" is just another way of saying "increasing forever" or "given a number x, you can always use x+1". This is why expressions like “infinity = infinity" or "infinity = infinity+1" like the comic are not useful statements.
That's also why your calculus professor is so insistent that you write out the whole limit notation, because it's nonsense to just throw infinity into an expression raw (like "infinity+1" in the comic). But, if you think of it as "the limit of x+1, where x approaches infinity", then it's clear that infinity doesn't have anything to do with the actual values, it's just used to describe potential values.
Here's an example if that still doesn't make sense: Bob and Jill are twins who were born with 0 and 1 dollars respectfully, but both earn a dollar a day forever because they're immortal. Just because they will live forever, doesn't mean that they'll ever be able to say "I'm infinite years old". They'll always be x years old, but x will increase by one every year from their birthday for the rest of time. For the same reasons, they'll never be able to say "I have infinite money", but if they don't spend it, it will increase forever, approaching infinity. And finally, if neither Bob or Jill spends anything and that dollar a day is their only income, then Jill will always be worth a dollar more than Bob, even though both have infinite wealth potential.