The Custom Lineage sidebar in Tasha's Cauldron of Everything states (p. 8; italics for emphasis mine):
Instead of choosing one of the game’s races for your character at 1st level, you can use the following traits to represent your character’s lineage, giving you full control over how your character’s origin shaped them:
Creature Type. You are a humanoid. You determine your appearance and whether you resemble any of your kin.
This indicates that, while a custom lineage character may look like a certain race, mechanically it is just a generic humanoid. This choice is made "instead of" one of the game's races.
Rules designer Jeremy Crawford has offered unofficial guidance on Twitter which reinforces this (hat tip to pdegan2814's answer for finding this):
In Tasha's Cauldron of Everything, the custom lineage option is chosen in lieu of a race, such as elf or dwarf. If you choose the custom lineage, you don't qualify for things in the game that require elf, dwarf, and the like.
It is important to realize, however, that race can be a sensitive topic for players and there's a risk of invoking things like Native American Blood Quantum laws or general "racial authenticity" in game which players may have real world experience with.
So, with the aim of preventing negative associations for the players, the DM ought to know his/her audience and be ready to also invoke DM fiat to allow exceptions to what the game mechanics indicate should happen.