Yes you can legally put your neighbors mail in their mailbox if it were accidentally delivered to your mailbox instead.
I've gotten some of my neighbors mail before and have put it in their mailbox and everything has been fine.
Now that I get my mail at the post office sometimes I still get a neighbors mail and what I do is just take it along with me and then deliver it directly to the neighbor when they get home.
So that way I'm sure they get the mail and it doesn't take that much more effort for me to just hold onto the neighbors mail and hand it to them when I see them the same day, evening or the next day.
Sometimes I'll knock on their door and let them know I got some of their mail and then hand them the mail and they are grateful that I just kept the mail and brought it to them so it didn't get further lost in the mail.
But if your neighbor has an outdoor mailbox or a mailbox on their house or apartment then yes you can open that mailbox and put mail in their that was delivered to your mailbox.