It's set that Jedi are not allowed to have children, but can they be considered godparents to someone else's child? True that this isn't a legal binding, but they're not allowed to have romantic relationships even outside of a legal marriage (which they also cannot do), so emotional and spiritual connections to other people are generally avoided.
Has this ever happened to a Jedi, blessed by the Jedi code or not?
Probably not.
Amongst other dictates, the Jedi Code forbade Jedi Knights and Jedi Masters from taking on more than one Padawan at a given time; and forbade Jedi from forming attachments, such as marriage, and other specific, individual bonds, such as family and romantic love.
Jedi Code | Wookieepedia (emphasis mine)
A godparent is a person who presents a child at baptism and promises to take responsibility for their religious education. One could argue that a Master/Padawan relationship is akin to this. But understanding the spirit of the question, the answer would be no, both because of the Code, as well as logistical issues.
We know that godparents do exist within the Star Wars main canon;
“Saw means something to a lot of people,” Maia said when Jyn didn’t answer her. “I heard about him as a hero of Onderon, from my godfather, Lux. Staven’s family was saved when Saw organized a supply run past Imperial blockades to feed them. Saw fought in the Clone Wars; he’s fought in battles since then. He’s half legend.”
Star Wars: Rebel Rising
But beyond that we don't really have enough information to judge. Jedi are certainly allowed to have friendships (Obi-Wan and Dexter Jettster, for example) and even non-romantic relationships but whether having a god-child would fall under the header of a
"possessive attachment"
isn't explained anywhere in films or books.
Within the EU canon, there's an example of a Dark Jedi (e.g. a Sith apprentice) who's a hold-parent (a term that is described as being analogous to a godparent) in Star Wars: The Dark Crystal. Lord Hethrir of Ferrerre kidnaps Leia's children and tells them that he's their hold-father.
I knew it!” Jaina said. “I knew Hethrir told a lie. He told lots of lies.”
“He’s a mean man,” Jacen said. “I don’t want him to be my hold-father!”
“He is not your hold-father, children,” Rillao said. “Is that everyone? No one left in the tree?”
Even though the answer is and should be no (as stated in amflares answer), it is at the same time yes. The yes comes from how the order works. It takes in young children and trains them and then before puperty they are given their jedi masters. They take care of them, train them further, ... . In essence so the function that godfathers have when the parents are dead.
Thus the Jedi / Padawan binding could be counted there as them being effectively the godfathers of their Padawans. And despite the code, all Jedi felt at least some bonds to their Padawans (at least how they were each portrayed in the series and films each).
So yes officially the answer is no, technically the Master/Padawan binding is almost the same though.
