To add on to J.R.'s answer: The problem here is, I think, the juxtaposition of "common" and "words". The most frequent and obvious interpretation of this pairing is ordinary words, not shared words.
Compare this with something like:
This anatomical structure has a common functionality between the two species, but the underlying physiological mechanism is very different.
Here the pairing of "common" and "function" is more likely to be interpreted as similar function.
This is probably why you've seen native speakers use the word "common" to mean "in common", but in these cases you have to be careful with your exact wording in order to avoid confusion.
Both English and German share a common vocabulary, although differences in pronunciation and conjugation sometimes make this difficult to hear in ordinary speech.