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In Game of Thrones S08E02, Jaime Lannister said,

“You don’t need a king. Any knight can make another knight".

But is it right in the context of Game of Thrones? Is it legal to make anyone else knight by a knight himself? Is it analogous to the real world?

Yes. In the shows this is the first time we've known rules about a knighthood been stated, so there's no other claim to compare it to. But, lucky for us, GRRM has stated it's either a knight or a King, but not a lord unless he's a Knight himself:

To settle an old debate on EZBoard, any king can make a knight but any lord cannot. That lord must be a knight as well. So Baelor I could make knights but Eddard could not. George said the more important thing for kings is making lords. The problem is giving lands.

In the books there is one notable example (spoiler alert, as this does not happen in the show):

Berric Dondarion knights Gendry

For the final part of your question: in real life there have been many types of knighthoods, some military, some just a title, some hereditary, ... . So there is not a fixed ruleset on becoming a knight. I'm not sure if GRRM based his knights on some real order, but it seems more likely he invented his own order for Westerosi knights and his rules are all that matter.