The player character is sometimes falling a small distance after entering nether or overworld through the portal (I believe it might be about 1 or less than 1 block). Sometimes you can even hear a falling sound effect, but so far it hasn't harmed the player character.
I tested it on portals with a different y coordinate (different altitude). I only used the smallest portals possible (the ones with 4x5 obsidian frame). I didn't test it on a pair of portals with the same y coordinate.
Steps to reproduce:
1. Create a pair of portals (matching x, z coordinate, different y coordinate)
2. Go through the portal
3. When you enter the other dimension, you fall a bit (you can see it with debug information turned on).