If I asked you to divide an apple by a stone, what would your answer be? Infinity is simply not a thing that your carefully-honed real-world intuitions about division apply to. Hilbert's hotel is precisely a thought experiment to prove this: divide an infinitely large set among infinitely many people, and you can get many different results. Any true statement about infinities you construct with division is true only by coincidence, not for the usual maths reason that things are true (namely that they are the result of following some broadly-applicable rules).
There is no number called "infinity", so there's no reason to expect the rules of the arithmetic of numbers to apply to it; you have to carefully go and prove that the rules you want to apply have some meaning when you extend the language in this way, and moreover that these rules are true. For example, there are multiple meaningful ways to add infinite quantities, e.g. depending on whether you are considering the cardinals or the ordinals. Those concepts are the same in finite-land, but are different in infinite-land. Cardinals do not admit the notion of "division"; ordinals at least have the division algorithm.
There is no number called "infinity", so there's no reason to expect the rules of the arithmetic of numbers to apply to it; you have to carefully go and prove that the rules you want to apply have some meaning when you extend the language in this way, and moreover that these rules are true. For example, there are multiple meaningful ways to add infinite quantities, e.g. depending on whether you are considering the cardinals or the ordinals. Those concepts are the same in finite-land, but are different in infinite-land. Cardinals do not admit the notion of "division"; ordinals at least have the division algorithm.