Can I use a monitor connected to an Xbox without an Xbox because it is still connected but then doesn't stop?
Yes you can
If the monitor has several connections and you have connected something else, you can also use the monitor without an Xbox
But I mean, if I only have the Xbox connected, I can watch YouTube, for example, because the TV can also work, right?
No, if the monitor is not a "Smart TV" you need a Fire TV Stick or something similar