Your 2021 MacBook M1 Pro, if bought new, will come with a pre-loaded version of iMovie 10 on it. iMovie is not part of the operating system but rather is a separate app that Apple provides free. If iMovie 10 is not on your new MacBook, you can download it for free from the online app store.
I am not sure whether your old iMovie 10.1.8 will run on Monterey. I suspect not. However, if you copy (copy, not move) your iMovie 10.1.8 library over to your new MacBookPro and open it with the newer iMovie version, the library will update and open, but no longer can be opened with the old 10.1.8 iMovie version. That's why you would want to copy of the old library on your old Mac instead of move the old library over to your new Mac, so you can keep a workable copy of the old iMovie library on your old Mac, that can still be opened with iMovie 10.1.8.
With the old iMovie library on your new Mac, there may be some issues with incompatible legacy media when you try to open it on your new mac Legacy media includes old formats, like .avi, that won't run on the newer iMovie versions after Mojave. The legacy media would need to be converted to a compatible format before it could run on iMovie on 64 bit operating systems after Mojave. With Mojave, you can do a File/Check Media Compatibility and be given an option to convert any legacy media that it finds. I don't know whether that option would be available on the old iMovie version that you have. I would guess that you are probably running Sierra or High Sierra.
In any event, as long as you keep a copy of the old library on your old Mac you would be safe. As mentioned above, when you place a copy of your old library on the new Mac, it will be upgraded when you open it with the new iMovie version. The old iMovie version will no longer open it unless you keep the original old library on your old Mac.
-- Rich