The name of the companion is made up of "Lovebirds", which is the name given to lovebirds and which means "birds of love" in French. Indeed, couples remain united all their lives and spend hours grooming each other and tenderly pecking ... If they are not in a relationship, they can live with a partner of the same sex. However, solitude does not suit them! They are unhappy and become garish because even if you devote time to them, you can never replace a feathered companion.
It could also be composed of "love" and "ignis" (meaning fire in Latin). The whole name would mean "fire of love".
The fact that they can cohabit with a partner of the same sex could explain why the Lovigis can be a female or a male when it is a pair of birds.
In episode 9 , when you have to choose who to settle in the Crystal Room when you have to chat about Yvoni , you start chatting with one of the boys (the one you chose). Then Miiko says to the caretaker and to Nevra/Valkyon/Ezarel (depending on the choice) "Are we bothering you lovigis?".
In episode 13, if we come back from the beach with Nevra, Ezarel says: "Ah, the lovigis are back."