When you construct numbers using sets under ZFC axioms or inside lambda calculus what you get is representation. But 1 is just 1.
1 ∈ Z and ASCII '1' can similarly be seen as corresponding in terms of having the same glyph when displayed. But of course, there are far fewer meaningful operations common to the two.
Saying that 0 belongs to 1 is false no matter what one uses to represent those numbers in any ZFC formalisation of numbers.
It’s a map-territory distinction.
On a more serious note, if you are of a certain philosophical bent you may believe that the natural numbers have an existence independent of and outside of the minds of humans. If so, 1 is presumably not a set, even if we don’t fully understand what it is. I certainly don’t think of it as a set on a day to day basis!
But others may deny that the territory even exists, that all we have are the maps. So in this one map, 1 is a set containing zero, but in that other map, it is something different. The fact that all the different maps correspond one-to-one is what counts in this worldview, and is what leads to the belief – whether an illusion or not – that the terrain does indeed exist. (And even the most hard nosed formalist will usually talk about the terrain as if it exists!)
But this is perhaps taking us a bit too far afield. It is fortunate that we can do mathematics without a clear understanding of what we talk about!
For that matter, how do we know what infinite sets like Z and Q 'literally are', without appealing to a system of axioms? The naive conception of sets runs headlong into Russell's paradox.