You didn't name the object when you said "let this thing be X"; you actually had already identified that "thing", and that process of identification was the process of naming it. You then defined some syntax ("X") and said that it was a name.

But there are things for which you can't even say "let 'this thing' be…". For example, ZF proves that there are uncountably many reals. There are only countably many names, so there must be unnameable reals. You can talk about "generic" reals (you can say "let x be a real" and do all sorts of interesting things with a generic x), but there are specific reals you will never be able to name specifically enough to distinguish them from their uncountably-many brethren. That doesn't make them "vague, undefined or ephemeral"! They're just so numerous that you can't describe the distinctions between them.

(Even hardcore constructivists usually accept enough Choice to prove the reals uncountable, although https://arxiv.org/abs/2404.01256 made headlines when it was shown not to be necessarily true.)