Take care can be used as a closing greeting that really cannot be picked apart further, a mere synonym of goodbye.
It's not the case, though, as you suspect that it "doesn't contain any other meaning", because it can also mean, well, take care. Subtle clues of intonation and body language can let someone know that when you say "Take care", you are trying to tell them that you are wishing them caution and strength as they face the challenges ahead of them.
In uses other than the two-word form, the words certainly have their core meaning: take care not to forget your passport or take care of Fido and Spike. This also goes for expanded forms of the parting words, such as take care of you[rself], you take care, or take care, out there, all of which more clearly genuinely communicate more explicitly the desire for the recipient to, well, take care.