Is there a grammar problem in this sentence"

Yes. The tenses of the verbs don't agree.
Either change got to get or don't to didn't (Not both).

I would write this as :

Native speakers grow so used to the figurative language that they speak, they don't realize how large a proportion of what they say and write is metaphorical.

Try to avoid get and got. They are overused and may not mean what you want to say. There is also some confusion in the sentence for clauses and phrases.

Without messing with what the author is writing, this would be acceptable and simpler. Retaining the "got" and "what" as a responsive would allow the sound of this sentence to fit in with whatever the author wishes to sound like. The hard job of doing editing is to correct any errors without changing the style of the rest of the work.

I would use "become" instead of "got" or "grow."

Note that the timestamps and dates of this thread are almost 2 years old. FWIW, it's doubtful the OP will be returning.

Does it matter?

Well said, Hermione.

Well, it maters if you want to provide an answer to the OP. One might doubt that there's another person who might want to know about that particular grammatical issue.