There is a long controversy as to what should count as the "size" of an infinite set, and there provably does not exist a notion that satisfies both the bijectivity principle, a.k.a. Hume's principle (bijective sets have equal size), and the part-whole principle (whole is greater than its part). So any notion of size for infinities will be counterintuitive in one way or the other. The history of philosophical debates surrounding this clash of intuitions is described by Mancosu in Measuring the size of infinite collections of natural numbers: was Cantor's theory of infinite number inevitable? (his answer is no).
Modern mathematics adopted the notion of cardinality that satisfies the bijectivity principle, but its downside are the "paradoxes of infinity" like the Hilbert's hotel that can accomodate any 10 (or 100, or 100000) additional guests at any time. It has rooms numbered by positive integers and to free up rooms 1 to 10 (or 100, or 100000) the manager needs only to move all existing guests from room n to room n+10 (n+100, n+100000).
The part-whole principle can be accomodated, but only if one gives up the bijectivity principle. One version was introduced by Katz, who shared the OP sentiment, in Sets and their Sizes:
"Cantor’s theory of cardinality violates common sense. It says, for example,
that all infinite sets of integers are the same size. This thesis criticizes the arguments for Cantor’s theory and presents an alternative. The alternative is based on a general theory, CS (for Class Size)... Because the language of CS is restricted... the notion of one-one correspondence cannot be expressed in this language, so Cantor’s definition of similarity will not be in CS, even though it is true for all finite sets."
Another alternative was introduced by Benci in 1995, and is called numerosity, "an Aristotelian notion of size", as he put it. Numerosity is always smaller for proper subsets, but... it depends on how a subset is given, it "counts" labeled subsets, not just subsets. So as long as a bijection changes the labeling too much it does not count. Bijective sets can, and do, have different numerosities. Mancosu uses numerosities to give an interesting counter to Gödel's argument that adoption of Cantor's cardinality was "inevitable":
"Gödel’s reflection aims at showing that in generalizing the notion of number from the finite to the infinite one inevitable ends up with the Cantorian notion of cardinal number. The key step in the argument is the premise and the theory of numerosities can help us see that the premise already contains in itself the Cantorian solution. In fact, the premise takes as evident the request that “the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (e.g., their colors or their distribution in space).” While the premise constitutes no problem when dealing with finite sets, one might question its acceptability in the realm of the infinite.
Indeed, in the theory of numerosities we cannot grant the premise when it comes to infinite sets. For, while it is possible to abstract from the nature of the objects themselves there is one type of relation that affects the counting, namely the way in which the elements are grouped. Such grouping makes no difference in the realm of finite sets of integers. But when we move to infinite sets a rearrangement of the grouping will in general affect the approximating functions and thus the numerosity of the set. Someone committed to the counting embodied in the theory of numerosities might thus reasonably resist accepting the premise on which Gödel bases his argument and thus also resist the claim that the generalization of number from the finite to the infinite must perforce end up with the notion of cardinal number."