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First of all categorical in a math context does not always refer to category theory. In logic a theory is called categorical in K if it has only one model of cardinality K up to isomorphism.

Second, is this really a common problem? People using the word categorical in a math context without mean either category-theoretical or the meaning in logic? Do you have an example?

I am same morphism on same object

Counterpoint: How many words are left that aren't overloaded in a mathematical context?

Assuming that by "word" you mean a finite string over a non-empty alphabet, then there are infinitely many - only a finite subset of them have mathematical definitions :) .

Would not every word also denote a regular expression, thereby having a mathematical definition?

CS is full of proofs that use all possible words or word sequence. The real test is if somebody actually wrote it in an undecipherable wikipedia page.