The bug

When playing on a server if you move vertically next to biome borders, IE between a landmass and water, the biome will shift between the land biome and the water biome. Even though the biome appears to have changed in the F3 menu it's generally a different value than what the client sees. If you generate the same seed in a single-player world then the biome doesn't change vertically and the biome borders behave as expected.

How to reproduce

1. Generate a world on a 1.16.5 server
2. Find a border between two biomes and stand on one side
3. Press F3 and move vertically to see the biome change
4. Copy the seed and generate a single-player world
5. Find the same location and check the biomes vertically
6. See that the biomes do not change

I noticed this when expanding a mushroom biome all the way to the edge and kept seeing mobs spawning where it should have been mushroom biome. So the server is still respecting what the world biome should be. I've also tried to load a world generated in single-player to multiplayer and experienced the same issue. What I was able to do was map out the biome border in single-player and use a mod to overlay the blocks on my server. The border that my single-player world biome border prevented all spawns on the extended biome even on the server. 

I've attached a video of me performing the reproduction steps between a river and a badlands biome. I've really only tested between water and land because I find it the easiest but I'd imagine it happens between all biomes.

Video demonstrating the issue: https://www.youtube.com/watch?v=iYhuhBmsgns