> I think you're mostly talking about orthography (writing), and not language itself. You might be saying that because of the different writing systems, the spoken language is different today, which seems plausible. In any case, no matter how "mixed" a language's history is, I bet that for its native speakers, it is acquired as one language and feels like one language. When people learn and acquire language, they don't have to trace any origins of any words.
I don't know how native speakers of other languages think about this, but for German (my mother tongue) I don't think this is true. The reason is that in German loanwords often keep their spelling. So if you want to know how to pronounce a loanword you better have a good intuition from which language it might come from and how it is pronounced in this language. By some "language nerds" in Germany it is even considered as a sign of education to apply the correct rules of the source language to form the "correct number" instead of the "Germanized number" when declining the word.
An example: The commonly used German word for "the courgette/zucchini" is "die Zucchini (singular)". And it is common usage to use "die Zucchinis" to form the plural.
Language nerds will disagree: In Italian, from which this word is loaned "zucchini" is already the plural, so the "correct" German word for this vegetable has to be "die Zucchino" and "die Zucchini" must only be used to refer to the plural. And then even more hardcore language nerds will come and and point out that "zucchino" is male in Italian, so it is wrong to make it female in German, so "der Zucchino" is the "ultimately correct" German noun for this vegetable.
I am really not kidding.
OK, for more sane examples:
You better know that "Giro" (as in "das Girokonto") or "das Cello" come from Italian - otherwise you will pronounce these German words wrongly. The same holds for "das Trottoir" (an old-fashioned German word for "the sidewalk"; the common German word is "der Bürgersteig") - you better know that this word comes from French and how French words are pronounced. I don't even want to start with English loanwords...