The answer lies in the difference between a cardinal and an ordinal. A cardinality is a size, an answer to "how many". As in 1, 2, 3... An ordinal is a position. As in first, second, third...
For finite numbers they are trivially equivalent. No so for infinite. If you add 1 to ω you haven't changed the cardinality at all. But if you append a new number after ω then that new number is in a position which doesn't exist anywhere in ω, and therefore it must be a larger ordinal.
So because cardinals and ordinals are different, there is no problem with ω+2 being a larger ordinal than ω, even though it has the same exact cardinality.
Also of some note is the fact that ω+1 has something that ω does not, it has a single unique element that is greater than all other elements, something that ω doesn't have.
No so for infinite. If you add 1 to ω you haven't changed the cardinality at all. But if you append a new number after ω then that new number is in a position which doesn't exist anywhere in ω, and therefore it must be a larger ordinal.
It's worth noting that ordinal addition and multiplication are not commutative. 1+ω = ω ≠ ω+1
For finite numbers they are trivially equivalent. No so for infinite. If you add 1 to ω you haven't changed the cardinality at all. But if you append a new number after ω then that new number is in a position which doesn't exist anywhere in ω, and therefore it must be a larger ordinal.
So because cardinals and ordinals are different, there is no problem with ω+2 being a larger ordinal than ω, even though it has the same exact cardinality.