(Java Minecraft 1.14) Does a scoreboard tag affect entity despawning?

The only NBT-tag that affects mob despawning is the PersistenceRequired-tag, a custom name does NOT change this behaviour.

This command summons a renamed zombie that will despawn:

/summon minecraft:zombie ~ ~ ~ {CustomName:'{"text":"zombie"}'}

This command summons a renamed zombie that will not despawn:

/summon minecraft:zombie ~ ~ ~ {CustomName:'{"text":"zombie"}',PersistenceRequired:1b}

Changing the name of a zombie with a name tag will change both the name and set the PersistenceRequired-tag to 1.

If a zombie picks up an item, then the PersistenceRequired-tag will be set to 1.
^{Credits go to Fabian Röling for mentioning this in the comments}

In conclusion, using this command will not make the nearest zombie despawn-proof:

/tag @e[type=minecraft:zombie,sort=nearest,limit=1] add zombie

Note: I only tested wether or not a zombie despawns in those cases, it may be possible that they take longer to despawn, or despawn quicker, or that they are affected in some other way.

Note 2: I only tested this with zombies, but it should apply to any entity