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.
It could also be composed of "love" in English and "ignis" in Latin, which means "fire". 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 Erika has a chat about Yvoni, she starts chatting with one of the boys. Miiko then arrives and asks: "Are we bothering you Lovigis?".
In Episode 13, if Erika comes back from the beach with Nevra, Ezarel says: "Ah, the Lovigis are back."