You must complete the Basic Challenge first in order to unlock Endless mode.
One of the hardest achievements of Lumines Remastered. The goal is to complete two rounds of the Endless Challenge. Not two skins, two laps. So for a lap to count, you must reach level 105 on "Lights" and return to "Shinin'" for a lap count, and all of this must be done in one go. Stamina and quick thinking is key here so playing the other modes can help you get acquainted before tackling this one.
You can earn the Golden Blocks achievement here as well and your reward is the lion avatar and you get the Rex avatar for completing two rounds of the Endless Challenge.
Video of Endless Challenge: