I don't actually think it is desirable to make mathematical notation unambiguous, if that is even well defined. To again use natural language as an analogy, there are only some 26 letters in most Latin based alphabets, but there are many more sounds. The same "letter", or historical sound, is pronounced differently in different contexts, but since it's almost always predictable based on context or knowledge, the system works.

In the same way, you could say that in one sense (though possibly not the mathematical sense) there are lots of different uses for "=". Most of the time this obviously hasn't been an issue, if it had been, people would have introduced new symbols. For some cases they have done just that, it can sometimes be useful to explicitly distinguish definitions with ":=" or a triangle above the =. In the same way people sometimes use a triple line = for "is identical with". But seeing that everyone knows these symbols, but still make do with just = in most cases, even they are only half useful.

If we use ordinary language as an analogy, we're usually all speaking the same one, and ambiguity is manageable.

Whereas in the professions, we are so deep into sub-specialties and sub-languages that even without ambiguity we are approaching a Babel where mathematician does not understand mathematician and lawyer does not understand lawyer.