What does 推し Oshi mean?

An oshi is the person you back. The word is built from a verb meaning to push forward or recommend, and the pushing is the substance of it: an oshi is someone whose success you are working on.

Where it came from

Idol fandom. The term spread through the audiences of large idol groups, where the structural fact that mattered was position: which members stood at the front, which appeared on the single, which got the camera. Fans did not merely like a member, they campaigned for one, and in some groups that campaigning was formalised into voting.

A vocabulary describing advocacy rather than preference makes sense in that setting, because preference on its own accomplished nothing.

Why favourite is the wrong translation

Favourite describes a ranking held privately. This describes a commitment acted on publicly: buying, promoting, recommending, showing up, telling people. The English word is a state and the Japanese one is closer to a role.

The nearest English equivalents come from other domains. Backing a horse, championing a writer, supporting a football club: each has the sense of a person whose fortunes you have attached yourself to. None of them is available for a character in a television series without sounding odd, which is the hole the loanword fills.

How far it has spread

Well beyond idols. It applies to anime characters, athletes, musicians, streamers, and in casual use to more or less anything a person is loudly enthusiastic about. Related vocabulary has grown around it for the activity of supporting one, which is itself evidence the concept was doing real work.

What the word assumes

That enthusiasm is a practice rather than a feeling. Under this vocabulary the question is not who you like but what you have done about it, and the answer is measured in attendance and spending as much as in affection.

That is a demanding way to organise fandom and it is not obviously healthier than the English arrangement. It is, though, an honest description of how a great deal of modern fan activity actually works, and English has no single word for it.