This is a very old consideration (cf https://www.jstor.org/stable/2183530).

If you want to frame a 'universal defining property', you have to frame it in some language. The definition, therefore, is inevitably idiosyncratic to that language, and, if you specify the same property using a different language, the definition is going to change. What's more, the two definitions can be incommensurable.

And---all you have done is translate from the object language to the meet language--leaving the definitions of the terms of the meta language completely unexplicated.

The solution to this is as old too---like "point", "line", etc, these terms can--and arguably should--remain completely un-defined.