In ME3MP, with the credits you accumulated from playing matches, you bought various types of packs. Each type of pack was intended to unlock progressively more rare items (although in practice you could get any rarity drop from any pack).
To unlock all the possible characters, weapons, weapon mods, and gear would take millions upon millions of credits. The randomness of the store made it a frequent target for hatred and for mock-serious rituals to appease the Random Number Generator/God (RNG).
In DAI MP, with the gold you accumulate from playing matches, you buy various sizes of chests: Small, Medium, or Large.
You can also buy a healing potion chest (which only contains healing potions), or a mixed potions chest (which contains some healing potions and some other potions). The potions chests are roughly analogous to ME3's Jumbo Equipment Pack in the sense that they only give you consumables, never permanent items.
The main difference in DAI MP is that, while there is still a rarity system, it is not tied to the type of chest. Any size chest (Small, Medium, Large) can drop any rarity, so the only reason to prefer one size over another is because of cost per item.
Current consensus is that Small chests are not a worthwhile investment because their cost per item (what is it exactly?) is considerably higher than that of Medium or Large chests, which have an almost identical cost per item (240 gold/item for Large, 241.7 gold/item for Medium).
The reason I stated that the DAI MP store is still obnoxious is because of how the rarity system interacts with the item level system.
Every piece of gear -- weapons, armor, accessories (rings, amulets, necklaces) -- can be Common, Rare, or Unique.
Every piece of gear also has a level, from level 1 up to level 23.
In general, higher level items are strictly better than lower level versions of the same item. This interacts… strangely with the rarity system. You can unlock a rare weapon whose level is much lower than the common weapon you're already using, in which case the rare weapon is almost entirely useless as an actual weapon for gameplay. (You can still salvage the obsolete rare weapon for crafting materials as explained in the next section.)
In addition, because of the item levels, it's never going to be possible to max out an unwanted weapon the way you could in ME3. In ME3, if you didn't like (say) the Eviscerator shotgun, you would eventually stop getting it from packs once it reached level 10. In DAI MP, it appears that no matter how many times you get (say) a Barbarian Maul, you will always have the potential to get another Barbarian Maul in future chests.
What's more, the restriction of weapons, armor, and (some) accessories to certain classes means that any given Rare or Unique you do get in a chest may or may not be something you can use for your favorite character. For example, only a mage can use a staff, so if you get the Level 23 Unique Staff of Awesome from a chest you can't use it with your favorite warrior or rogue. This is in contrast to ME3, where any weapon could be used by any character.
In short, the store in DAI MP feels even more obnoxious than the already obnoxious store in ME3 MP.