In many cases there is a faster journey possible by leaving an overnight train at an intermediate stop and taking a faster day train, this is what the CD site is showing you.

The sites do not know that you want the direct train with no changes and will default to the fastest possible journey, change the settings to show only direct trains.

If you look the interrail planner also shows this option above the direct train you have selected.

Thank you so much. Now I try to reserve beds for that train on the Nightjet website and it tells me “no offers available” (screenshot 1). I have tried it on the cd website and it does not allow me either (screenshot 2). I have written to CD and they have told me that I can only book in person at the station (screenshot 3). How is it possible? Obviously I can't go to Budapest to reserve the seats. And if I wait until I'm there, I run the risk that there won't be any more beds. What can I do?

Have you tried to make a reservation through ÖBB?

That EN train is normally available to book through https://shop.oebbtickets.at/en however it is not currently available for that date or a few weeks either side. There is an alert of changed times due to construction work, this may mean the times are not yet finalised and it will become available to book at a later date or it may not run as planned at all.

You could contact OBB and see if they can explain the situation.

I have tried seat reservation by shop.oebbtickets.at and it shows two possibilities, one with a connection and one direct (screenshot 1). The direct one doesn’t allow to reserve tickets. The other one warns you that there is no service available from Budapest to Ulm, what is basically all the route! (screenshot 2). All in all it’s a bit worrying and uncertain… moreover travelling with kids.

1. Do NOT use "Seat reservation only” but add the Interrail/Eurail discount and then select "One-way tickets and day tickets”.
2. Reservations from Budapest are not yet possible on that date. I don't know whether that is because of engineering works and the timetable not being final yet, or because reservations from/to Hungary only open later (currently they're possible until 11 August).

I'd say wait 6 days and then check again.