Same in french, in this case I think it implies that God is "part of the family" so to speak, someone very close, not a stranger.

"Vous" is formal but it doesn't necessarily mean that "tu" is informal, it just implies a certain form of intimacy.

Also, I'd like to point out that using the plural "you" as polite/formal also kind of works with plural "we" as in the "royal we": https://en.wikipedia.org/wiki/Royal_we

I'm not sure if both are directly related though.

Biblical Greek has no formal, only singular/plural. Same with vulgar Latin. Latin and Greek are rude languages and the our father is a rude prayer, so tu/toi/thou are perfectly appropriate.