What does Magic: The Gathering have that is analogous to Yu-Gi-Oh!’s level system?

It isn't much of an analogue but you could say 'converted mana cost'.

A YuGiOh monster's level usually determines how much game resource you have to expend to put that card into play. So a monster of Lv7 or more needs two tributes, or if it has a special summon method, will certainly require some other cost.

MtG has it much simpler - the more effective a card is (in general..) the higher the mana cost is to use it.

But YGO levels have a bunch of other uses for which there is no MtG analogue, also in general your spells/traps have no real 'cost', but in MtG non-creature spells share the same cost system as creature spells.

Also MtG mana is fundamentally different from anything YuGiOh has, being that it:

A - is generated from cards that have to be included in deck,

B - is expended temporarily and reset each turn,

C - has to be built up over time (meaning you can't play the big cards early unlike YuGiOh, but can almost always play resource heavy cards when you draw them during the endgame, where in YuGiOh you may have nothing left to expend)

D - comes in different types requiring some thought to be put into deckbuilding, a card that requires one G and four of any mana is easier to splash into a non-green deck via land and cards that produce multiple types, where one that costs GGG and two of any mana would probably only work in a full or half green deck.

Both card games use "costs".