As for parts of speechesspeech, I am quite sure it is not the case.
For instance, some languages are problematic in separating clearly verbs from adjectives like Japanese and Korean, some native American languages seem to have much less categories than Indo-European. Since Since each part of speech corresponds withto a phrase, the phrases can't be universal either.
Now sentence partsgrammatical relations are like roles certain phrases can take, and it strikes me intuitively, that these roles are limited by logic and can't allow for idiosyncrasies like partparts of speechesspeech can.
But is it really so, are not even some sentence parts universal among languages but each language has the same set of them?