“And” is a conjunction; one does not normally start a sentence with a conjunction. This construction implies that it is continuing a prior sentence or fragment.
“also” is not a conjuction, it’s an adjective on the object, indicating plural object, specifically an unspecified additional object.
“with you” is a preposition and object of said preposition. It is a dependent clause, not a sentence.
It is as if there is a third party speaking:
To the people: The Lord be with you …
then, to the celebrant: … and also with you.
The answer “And also with you.” is bad not becuase of the 1st two words, but because it is not a sentence. To make a sentence out of it requires additions and a deletion: “-]And/-] May it also be with you.” But that’s even FURTHER from the latin…
To the people: Dominus vobiscum … (The Lord be with you)
then, to the celebrant: … et cum spirtu tua. (and with spirit yours).
Again, a third party type dialog.
It’s a construct that is gramatically poor in either language, but conveys a specific meaning despite its grammatical idiosyncrasies.