can i use the new i mac as a second screen for my mac book pro?

No, Macs are computers they are not external displays for other devices. Please read Target Display Mode.

If your MBP needs an external display, please buy one. This is the most reliable, easiest and least expensive (does not require very expensive cables) method. You can easily find external displays for about $100-$150.

That would be very silly. Why have two Macs if they can't even talk with each other and cooperate or complement each other.

This can't be right. You can connect the two screens, via the apple menu (in the upper right corner, it looks like two batterylife icons on top of each other).

But there is one strange thing: if I mirror the screen of either computer - either one shows only a very tiny version of the other's screen. So they can cooperate, but apparently not that well (or I am doing something wrong - but then I find it strange that I don't get a notification or suggestion on how to do it properly.

Don't believe me then please read Use your iMac as a display with target display mode - Apple Support and Apple very clearly lays out what will work. TDM is dead.

If you run Monterey on them you can AirPlay to the new iMac. Otherwise, no.