Most of these are kinda loose but here you go.
Totsukuni no Shoujo: A tall, shadowy creature protects a young girl from the inhabitants of the cursed land she was exiled to. The Author also has several one-shots with monsters in the main cast.
Knights of Sidonia: The MC may or may not be considered a monster, but one of the main cast is a giant hybrid monstrosity (she cute tho) bred from man and giant eldritch space abomination.
Ogeha: The female lead is an alien-butterfly-monster the lead prematurely ripped from her chrysalis.
Parasyte: MC's arm is replaced with a sentient monster who intended to claim his head, and must work with it to avoid others of it's kind
Honorable Mention
Angel Densetsu, everyone treats him like a monster, and he has the face of one