The difference between "lift" and "carry" is that "carry" implies you travelled with the item after you lifted it. For example, you can stand on the spot and "lift weights", but if you are "carrying a weight" it means you are walking around with it. "Lift up" idiomatically means to pick something up. Some might say this is a tautology, but "lift down" idiomatically means to move something to a lower position.
Although you don't have to specify a direction when you use "carry" (it may be superfluous to what you are saying), you shouldn't be surprised when you do see a direction. In your example, that direction is "up".
Did you carry the bike up with you?
This is asking if someone carried the bike from one place to another in an "upwardly" direction. Quite what they mean by "up" is an entirely different question. It could literally mean up a flight of stairs, but English speakers also use "up" to mean relative compass directions, for example when you travel from a place in the south to somewhere in the north. This has been discussed previously on ELL.