This game popped up in the sales. https://www.gog.com/game/raiden_3_digital_edition
As you can see it would work better with the monitor held vertically which is one of the few advantages of the widescreen format over the old 4:3 or 5:4 shapes.
Are there any basic stands for screens that let you flip them around easily and are there games other than old school spaceship shooters that work well with this configuration?

It would still be better with 4:3, or more precisely 3:4. With a 16:9 monitor flipped vertically, you will have lots of space above and below the game. Here's a vertical 3:4 monitor for a mere $2,500, probably costs that much because they put "medical" in the name.

I don't know about monitor stands but I use pneumatic monitor arms for a few of my displays. They can be freely moved up, down, left right, tilted, pivoted and rotated. You might want to check some of those out.

I have a Dell monitor that can be rotated to portrait mode, I think this is common for Dell monitors. I have not played a video game this way, I only used it to read few comics.

There's something even better: monitors with pivot function.

These can be turned from horizontal alignment to vertical alignment.

Some examples: https://www.coolblue.nl/en/monitors/with-pivot-function?page=5

This looks the best at the moment unless you can buy an adapter to stick in the back of a monitor so it's like the products BreOl72 suggested without having to get a new monitor. You would use software or the right kind of games to fill the picture out properly.
For windows go to control panel display settings and switch it to portrait mode so you can see it.

I don't see it.

If your monitor is tall enough to be comfortable at the distance you view it from, then it does not matter how much extra space it has on the sides. 4:3, 16:9, 200:1, whatever, you can ignore all that space on the sides.

If your monitor isn't tall enough to game, then rotating it might be a nice workaround but I'd just buy a bigger monitor.

Fwiw: I have a 16:10 Eizo monitor that rotates, and it just gets too tall if I do so. I don't see the point of it in normal use in a fixed location where the distance between monitor and seat is relatively constant.

Don't forget that for using a monitor arm, the monitor must be VESA compatible.

If by "software" you mean stretching the image, no, that's not really acceptable. What sort of vertical games have a taller than 3:4 aspect ratio? Out of curiosity. I've never seen any. As far as I know all vertical games were originally old arcade games made by taking a standard 4:3 CRT and just rotating it 90° (hence the vertical scan lines).

So long as you have screw hole mounts on the back of your display, you should be able to attach an arm to it. Mine all additionally came with extenders, spacers and washers that allow you to attach them to a wider range of displays.

This is the one attached to the display (81cm/32in) I am using to type this post on:

Suptek MD5211B

It's no longer in stock, but they have what looks like a newer version:

Suptek MD5411B

Obviously there are many more brands and models out there, this is just what I know and have used for around three years without any problems. Hope this helps.

Recently, I have been looking at monitors in the hope of replacing my old one. Well it turns out that unless you want a shitty TN panel, all the models are curved(at least here in Romania). I'm looking for a proper FreeSync, VA panel, 144Hz, 1080p monitor.

https://www.ign.com/articles/the-best-1080p-144hz-gaming-monitor-2

Thanks for that, but your link just proves my point: flat screen are TN panels, while VA panels are curved.

What's the problem with curved monitors?

https://www.pcgamer.com/best-gaming-monitor/

And IPS monitors are what?