Nobody would ever say "She speaks Romanian NATIVELY" in any of the anglophone countries. They would say: "She speaks it like a native," or "She is a native speaker" or "she speaks it fluently" NEVER she speaks it NATIVELY!!! Never ..... not in a million years!!
So you know how in Latin a adjective modifying a subject can be translated with adverbial force? Is that essentially what's happening here with 'nativ'? Seems like it, but then it should be 'nativa', right? Hmmm...
What's happening is that "nativ" is used as an adverb. The DEX does not classify it as an adverb, so this might not be a correct, by-the-book thing to do, but it makes sense for such an adverb to exist; after all, English has "natively".
A lot of adjectives are used as adverbs even though the dictionary doesn't mark them as such. This comes natural because most adverbs share the form of a masculine adjective anyway (that's why it's "nativ", not "nativă"). If you want, you can always use "în mod ..." to keep things as adjectives:
Ea vorbește în mod nativ limba română. - She speaks Romanian in a native way.
But here the whole "în mod nativ" really functions as an adverb, so why not skip the extra words? Personally, I see it as a failure on the DEX's side to keep up with actual usage.
I don't know Latin, but I hope this answers your question.
But if we just re-invent language how we like, no-one will know what rules govern usage, chaos will ensue, and communication breaks down.