This is so in the Catholic Church as well.
Well, yes and no. I’m not sure how the Eastern Catholics do it, but Rome has a step-by-step process.
First of all, there are a lot of people in the Latin Church who probably “already are saints” but won’t be officially recognized or publicly venerated for one reason or another. Thomas a Kempis, the author of “Imitation of Christ”, is a prime example.
Second, what you’re referring to as “popular demand” is called “evidence of cult” in the Vatican sainthood process. It is one factor that the Church takes into account, however it is not the only factor. And probably with good reason, as sometimes people can get carried away venerating somebody who has questionable aspects for official sainthood. St. Simon of Trent would be a good example of this: since he was a very young boy (about 2 or 3 years old) when he died, he is very likely in Heaven because he was presumably baptized and also well below the age of reason for committing any mortal sin, so there wouldn’t be a basis for him to go elsewhere. So technically, yes the kid is probably a “saint”. But the public veneration of him was driven by antisemitism. The Church doesn’t want saints that raise those kinds of issues to be on the official list of saints.
The Vatican process would also look at the person’s life, their writings, any phenomena regarding their deceased body, whether they had been martyred for the Faith vs. dying in some other way, and of course, miracles. In recent years, some saints who have contributed in a major way to the Church, such as to theology, have been canonized without miracles, which to me is fine once in a while but I would not want it to become the norm.