If Leyline of Singularity is in play, will my opponent be able to have multiple "2/2 zombie tokens" in play?
It does count for tokens; you will not be able to have multiple tokens with the same name. The name of a token is defined by the spell that created it:
So in your example, if an effect creates a 2/2 Zombie token, then the name of that token will be "Zombie."
Furthermore, the legendary rule does apply to tokens:
A permanent is a card or token on the battlefield.[CR 110.1] If you have tokens on the battlefield, Leyline of Singularity will make them Legendary.
The legendary rule affects permanents, so it does affect tokens.[CR 704.5k]
The token's type is irrelevant. The legendary rule compares names.[CR 704.5k] However, a token's name is usually the same as the type as which it was created. Some effects that create tokens give them a name (e.g. Cloudseeder).[CR 110.5c] For the rest, the token's name is the same as its creature type. For example, the token created by Vile Rebirth is named "zombie".
If your opponent's 2/2 zombie tokens are named "zombie", they will have to put all but one in their graveyard.