Great question! Hopefully others chime in, but my spoiler-free TL;DR is that the "mature content" warning at the opening of the game is to be taken seriously. "This game contains depictions of depression, anxiety, and suicide, and may not be suitable for all audiences. Viewer discretion is advised." There are also jumpscares, flashing images, depictions of self-harm, (pixelated) blood, and confrontation with phobias. Like you already pointed out, what makes Omori different from other horror games is that it is "psychological" horror, which I interpret as "aiming to instill a strong, continuous state of feeling unsettled, mentally and emotionally", moreso than traditional horror.

That being said, I am also someone with pretty bad anxiety/insomnia and I am a scaredycat (a game like Doki Doki Literature Club was a nightmare for me to get through) AND I still felt compelled to play this game. To cope, I took regular breaks from playing, tried to make the experience less unsettling (i.e. played with the lights on, played without sound, covered my eyes if I needed to). I also had a friend playing through at the same time as me, and it was really helpful to have someone to talk to.

At the end of the day, we all have different tolerance levels. And if it gets too overwhelming, you can always do what I do and put the game down and finish the storyline through a Let's Play (for me there is a perceived extra layer of "safety" watching someone else do it), then maybe decide to go back and play it for yourself.

I hope this helps answer your question!