Minecraft 1.7.2, Minecraft 13w48b, Minecraft 1.7.4, Minecraft 14w06b, Minecraft 14w07a, Minecraft 1.7.9, Minecraft 14w20b, Minecraft 14w21b, Minecraft 1.7.10-pre4, Minecraft 14w25b, Minecraft 14w26b, Minecraft 14w26c, Minecraft 1.7.10, Minecraft 1.8-pre3, Minecraft 1.8, Minecraft 1.8.6, Minecraft 1.8.9, Minecraft 1.9 Pre-Release 3, Minecraft 1.9 Pre-Release 4, Minecraft 1.9, Minecraft 16w20a, Minecraft 16w21a, Minecraft 16w21b, Minecraft 1.10.2, Minecraft 16w43a, Minecraft 16w44a, Minecraft 1.11 Pre-Release 1, Minecraft 1.11.2, Minecraft 17w06a, Minecraft 1.12.1
As can hopefully be seen in my screenshots, obvious straight lines can be seen through the stained/hardened clay.
This may be only in mesa plateau, or that may simply be the easiest place to see it as it is more flat. The seed I used was "mesa" for those examples, but I've tried several seeds and found signs of the same thing in every mesa I've visited.
Sometimes it can be less obvious due to less contrasting colors at the surface level. As can be seen from mesa2, where I've put wool to indicate the chunk boundaries, these lines are chunk related. It seems that there's a chunk-sized area where one "height" of colors meets another (strip going north-south) and in this strip the "sides" alternate every chunk, producing these "fingers" going east-west. On one side of the "strip" the colors are assigned by Y one off from the other side. In mesa3 and mesa4 you can see that the effect isn't just on the surface, and goes all the way down in the clay.
Based on 1.11.2 decompiled using MCP 9.35 rc1
It looks like the method net.minecraft.world.biome.BiomeMesa.getBand(int, int, int) is causing this by offsetting the used clay band index.