This is a very deep and serious issue. There are some fundamental questions to ask about infinities, which may elucidate some deeper properties.
Take a simple infinity like that of integer number. As is a single entity, it must contain 'All Integer Number'. Note that this is a descriptor of a total unified set of a particular 'quality' of number. In this state of a single entity of the 'infinity of all integer number', viewed from the outside as it were, there is no particular (as in 'particles') structure where individual numbers are visible. This is the non-computational reality of the infinity.
And this is the source of this particular infinity paradox. If ALL integer numbers are there, then the structure is fixed, as a dimension-like [realm of quality of all integer number]. So on the one hand we have a unified infinite realm, a single entity of infinity, where the infinity can only be unified if all interger numbers are viewed as a 'whole'. While on the other viewing its 'contents' as individual particles of number we see individual particulate numbers, but an infinite number of them, which again is paradoxical. Interestingly there seems to be something of a 'wave/particle duality' paradox in this view of infinity.
The Hilbert Hotel makes a different interpretation of the infinity boundary viewed from the inside at the particulate level it assumes that because there is no limit on magnitude you can keep adding numbers infinitely - this is a local view with an indeterminate, infinite, boundary. Whereas the dimension-like view is to regard the infinity of integer number as a 'whole', a 'fixed', non-computational dimension-like unified realm of quality of all integer number.
The description of integer infinity that I gave above creates an interesting view. The infinity of integer number, as a dimension-like entity represents a change of state, a change of phase from the view of particulate individual numbers to a unified whole. This infinity refers to a 'realm of quality' and is a non-calculable entity, whereas the particles of number viewed inside are calculable, yet infinite.