multiple textures on one mesh + different Albedos

I'm currently trying to write my own minecraft clone. Just for fun.

I'm currently creating one mesh per chunk (16x16x16 blocks). I've worked with a simple sprite on a material with a normal sprite texture. Just setting the uv's while creating the mesh and I'm done. Easy.

Modern minecraft is using a system where textures can be set to different colors (albedos). For example a black/white grass texture which is colored depending on the current biome. (ie forest = #0b2c04 hills = #12E14F00)

I could set the albedo color for the whole sprite - but not for a single tile. And especially not on the mesh itself: I can't set it to bright green on 0,0,0 and dark green on 0,0,1

Now I could create a mesh per block type per chunk - which would decrease performance by a lot eventually. It also wouldn't solve the problem to set color per face.

So... I'm running out of ideas and wanted to ask your for some help.

Actual Question:

How do I set different textures on one mesh with different colors (albedos) per face? If that is not possible at all - do you have other ideas to solve the problem?