Series about a young man struggling to become king in middle ages setting

Perhaps it is the series beginning with The Prince in Waiting?

A thirteen-year-old’s expectations of royalty give way to adventure in the first book in the post-apocalyptic Sword of the Spirits trilogy from the author of The Tripods series.

In Winchester, roles are clearly defined. Warriors fight battles every spring. Dwarfs make the swords and the shields. Grotesque mutants are the servant class. Seers interpret the wishes and predictions of the spirits. And the Prince is the ruler of the city.

Thirteen-year-old Luke has no reason to suspect that any of this will change. It’s been this way for centuries...at least since the year 2000.

It was originally printed in about 1970, and reprinted recently, perhaps also in the 90s when you read it?

Possibly one of Katherine Kurtz's Deryni trilogies? There were several (and some individual books as well). Many of them deal with the early years of King Kelson's reign.

young man who becomes a king

Kelson became king in the first book of the first trilogy, Deryni Rising, at the age of 14.

in a medieval setting,

The country is Gwynedd. King, dukes, earls, barons, fealty, garderobes. Very medieval

lots of drama and backstabbing.

Check. He has to fight for the throne. Infighting. Traitors.

I can remember that it was part of a series, most likely a trilogy.

The Deryni series is made up of many books (the majority, I believe, published before 1990, though definitely not all). Within the larger series, there are several sets of trilogies, two of which are about King Kelson as a young man. The first, when he comes to the throne, consists of Deryni Rising, Deryni Checkmate, and High Deryni. The second, several years into King Kelson's reign, The Bishop's Heir, The King's Justice, and The Quest for Saint Camber.

fictional Europe or just some unnamed land.

Gwynedd; fictional land with fictional neighbors, with hints of medieval England and neighbors.

I don’t remember there being any magic or fantasy elements.

The OP also says, however,

Although I don't remember it having magic or fantasy elements, it definitely was either alternate history or just an non-Earth world. It's possible that it was a low-magic setting.

Here it is only a partial match. I don't know whether you can call this a low-magic setting. However, it could be a match anyway; the magic in this world is not witches or druids or that sort of magic, and 'spells' are only a small part and rarely mentioned. The Deryni are a subset of humans who have special abilities which include telepathy, telekinesis, and other more 'magical' seeming abilities. The background 'theme' running through all the books is the issues between the Deryni, who have powers, and the other people, who are the majority, many of whom fear and hate them.

Also I feel like this book was written in the early 90s as I remembering reading the novel as a brand new book in 91/92.

The first trilogy was published in the 1970's, but was republished every few years until the mid-1980's, with the third book, High Deryni, having a reprint (in the UK) in 1990.

The second was published in the mid- to late 1980's, in several reprints.

In most of the novel the main protagonist is a young man.

Check, Kelson, in the two trilogies I mentioned (there is another series about an earlier era; while there are important characters who are young men, the main protagonist, Camber of Culdi, is an older man).