#A variant human's starting feat and extra skill proficiency are racial traits, and are lost when reincarnated into another race.

The reincarnate spell description states, in part:

The reincarnated creature recalls its former life and experiences. It retains the capabilities it had in its original form, except it exchanges its original race for the new one and changes its racial traits accordingly.

The variant human's racial traits are listed as:

[...] your Dungeon Master might allow these variant traits, all of which replace the human’s Ability Score Increase trait.

 

Ability Score Increase. Two different ability scores of your choice increase by 1.

 

Skills. You gain proficiency in one skill of your choice.

 

Feat. You gain one feat of your choice.

These traits - along with the other base human traits - are all replaced by those of the new race.

The same is true of half-elves, who get the Skill Versatility trait:

You gain proficiency in two skills of your choice.

If the half-elf is then affected by reincarnate, they will lose these two skill proficiencies. The skill proficiencies come from a trait granted to the character by their race.

This question is not officially addressed in the Sage Advice Compendium, but 5e rules designer Jeremy Crawford unofficially confirms this interpretation on Twitter:

If a human is reincarnated, do they lose their bonus feat and skills?

 

If the reincarnate spell changes your race, you lose all your original racial traits and gain new ones.

That said, that is simply how the spell interacts with racial traits mechanically. However, some racial traits seem to reflect experience or knowledge rather than genetic/anatomical capability. In such cases, the DM might reasonably house-rule that the character can still access the trait or benefit since only their physiology is changed, not their memories and experiences. There's no official guidance for this sort of thing, so I would simply warn DMs to be careful about allowing this.

A variant human's starting feat and extra skill proficiency are racial traits, and are lost when reincarnated into another race.

The reincarnate spell description states, in part:

The reincarnated creature recalls its former life and experiences. It retains the capabilities it had in its original form, except it exchanges its original race for the new one and changes its racial traits accordingly.

The variant human's racial traits are listed as:

[...] your Dungeon Master might allow these variant traits, all of which replace the human’s Ability Score Increase trait.

Ability Score Increase. Two different ability scores of your choice increase by 1.

Skills. You gain proficiency in one skill of your choice.

Feat. You gain one feat of your choice.

These traits - along with the other base human traits - are all replaced by those of the new race.

The same is true of half-elves, who get the Skill Versatility trait:

You gain proficiency in two skills of your choice.

If the half-elf is then affected by reincarnate, they will lose these two skill proficiencies. The skill proficiencies come from a trait granted to the character by their race.

This question is not officially addressed in the Sage Advice Compendium, but 5e rules designer Jeremy Crawford unofficially confirms this interpretation on Twitter:

If a human is reincarnated, do they lose their bonus feat and skills?

If the reincarnate spell changes your race, you lose all your original racial traits and gain new ones.

That said, that is simply how the spell interacts with racial traits mechanically. However, some racial traits seem to reflect experience or knowledge rather than genetic/anatomical capability. In such cases, the DM might reasonably house-rule that the character can still access the trait or benefit since only their physiology is changed, not their memories and experiences. There's no official guidance for this sort of thing, so I would simply warn DMs to be careful about allowing this.