No. The game doesn't get that fiddly
Characters and their carrying capacity, and lifting capacity, is as granular1 and simulationist as the game gets. The Encumbrance rules are a variant/optional rule. (The campaigns I am in use them, but not all tables do).
Adding the problem of volume/cube for characters creates a significant bookkeeping and administrative burden that the rules' design intent wasn't trying to achieve. It would be a best practice to proceed thusly: if a particularly awkward situation arises in terms of the geometry of something to be carried, and how to carry it, it should be addressed by a GM ruling and/or player problem solving.
A baseline axiom for D&D 5e is Rulings over Rules.
Now that you edited your question with more detail ...
Is volume tracked per storage item or all as a whole (I mean, if I have an item which size 2 ft3 and two bags with 1 ft3 each one, can I carry that item or not?)?
How important is this to game play?
If you are a player: consult with your DM.
If you are the DM: does this item, and how it is carried, matter
enough that you want the players to have to solve that particular
problem? If so, pose that problem to them for them to solve. They may surprise you.
1 by "granular" I mean "does not go to that level of detail, or leaves it as an abstraction."
Why you might not want to add that fiddly bit: a PC can be as small as a halfling or as large as a goliath. The designers chose not to make that variance in size and mass have many effects, beyond small creatures having disadvantage in using heavy weapons. That's about as far into versimilitude as they went, although halflings can hide behind creatures one size larger. Note that both of those design decisions don't add extra bookkeeping or additional arithmetic.