Yes, it can mean I must date him, and that was my first thought , but it's not clear cut, it's going to depend on the context, less ambiguous are:
I need to go outside with him
I need to leave with him
In Portuguese outside of Brazil (Europe, Asia, Africa) and it turns out, some parts of Brazil too, then precisar de is always used even with other verbs and not just nouns.
This is okay on many of the exercises here on Duo as it has been expressed by several of the mods and course contributors that Portuguese from all varieties needs to be accepted, so if you encounter an exercise that does not accept "de" with precisar then please report it as it helps everyone. :)
Luckily Duolingo recognises this now! I just talked to my Brasilian boyfriend how annoying it is to remember about "precisar de" and just "precisar" with verbs but he said its absolutely correct if you just use "de" with "precisar" all the time :) We live in São Paulo state by the way.
"sair" derives from Latin "salire" meaning "to jump". But this is rather confusing since in the other Romance languages, this verb has different meanings.
Spanish "salir = to go out" (like Portuguese)
Italian "salire = to go up; to get in (a car e. g.)"
French "salir = to dirty" (though this may have a different origin, see salirophilia)
The Italian meaning "to go up" seems to be the most similar to the original Latin meaning "to jump".