What do you mean?
0 'zero' and
O 'oh' are not the same character, either historically (the characters have distinct origins - with the Roman letter
O 'oh' coming from Greek omicron, itself coming from Phoenician ʿayin; and the numeric character
0 'zero' coming originally from a dot notation in India (so dotted-zeros, which you mention using and I also prefer, are nicely appropriate) which eventually 'widened' to a hollow shape, adopted by Arabs and transmitted by them to Europe.), nor in terms of their encoding. But this isn't at all an ASCII innovation: the symbols were historically distinct and in fact have completely different origins.
So 0 'zero' and O 'oh' are not the same symbol. A is really the same symbol, whether in alphabetic use or hexadecimal, but 'means' different things in different contexts. (Even the alphabetic 'meanings' of A/a are not very consistent in English, in terms of representing sounds, and are influenced by context, so that a in cat and a in father don't 'mean' the same thing, in terms of representing sounds.)
To be fair, OP probably isn't actually using the Scandinavian Ø in contexts where it would be confused with ∅ (slashed zero), though in theory I suppose some language might support it as part of variable names etc.
But certainly I get irritated with fonts in which I (capital 'i') and l (lowercase 'L') look identical, as they occur in similar contexts and so increase cognitive load.
[Postscript edit: I was thinking that there is no visually similar character to a dotted zero, but then I remembered what upper-case theta (θ) is Θ, which does look awfully like a dotted zero.]