So if your playing on a multiplayer server with 3-4 people and you make a mob spawner, the mob spawner won't work that well because mobs will spawn near other players but what if everyone leaves and you're the only one there? Would the mob spawner spawn rate be the same as a single player spawn rate? Does it depend on the server?
While the mobcap for regular spawning depends on the number of players, it does not affect spawners.
MaxNearbyEntities: Overrides the maximum number of nearby (within a box of spawnrange*2+1 x spawnrange*2+1 x 8 centered around the spawner block) entities whose IDs match this spawner's entity ID. Note that this is relative to a mob's hitbox, not their physical position. Also note that all entities within all chunk sections (16x16x16 cubes) overlapped by this box are tested for their ID and hitbox overlap, rather than just entities which are within the box, meaning a large amount of entities outside the box (or within it, of course) can cause substantial lag.
This means that the number of e.g. zombies around a zombie spawner is fixed for that spawner. Even if you bring in zombies that were not spawned by that spawner, they still decrease the amount of zombies that can spawn there. And also it doesn't matter how many players are there, as long as at least one stands inside the activation radius.
I wasn't able to figure out just from research whether the mob spawner counts renamed mobs, too, but from that wiki quote it seems like it.