I read that you can take a night train from Belgrade to Zagreb (and I confirmed it in the DB website), but it first goes to Budapest, transfers, and then goes to Zagreb. I also know there's a direct train but I rather want to take the night train in order to save on the accommodation cost.
In this case, is the transit in Budapest handled as a one-day stay in the Schengen area? Or is it like a flight and there's no need to get through immigration?
Yes, this counts as entering the Schengen area. If you will be in Hungary at midnight, it will even count as two days.
With trains, you're generally processed into and out of each country when you actually cross the border. There are some exceptions, but the trains you are contemplating are not among them.
The concept of transiting a country's soil without entering the country only exists at airports, with only a couple of provisions for such transit by land transport in the world.
So yes, you'll obviously enter the Schengen Area, with border control taking place at the border.
