Chris is right, normally rubbing strips are held on with VERY strong double sided tape. Other cars sometimes have plastic trim clips held to the bodywork with rivets, but I don't think there are any on our Unos.
The problem comes when you try and remove the double sided tape as it tends to pull off paint if you're not careful. Sometimes if the trim has been removed and you want to remove the remains of the tape still stuck on the bodywork you can try the following:
Try soaking the tape in Methylated Spirits for a while using a rag, then using a plastic scraper (like a credit card) gently scrape from one end. If this doesn't work, try heating it up with a hairdryer (making sure any meths has evaporated as it is very inflammable!) which might soften the tape up to allow it to be removed. Again, be careful as it might soften the paint as well making it come off with the tape.
If you want to stick the trim back on, try and find some double sided tape from a motor factor. There are some tape products specifically designed for attaching trim to cars. You might also be able to find some glues also suitable. Make sure all traces of tape/ glue have been removed from both the bodywork AND trim before attempting to re-stick the trim pieces.
WARNING! DON'T use 'EVOSTICK' under ANY circumstances! EVOSTICK contains solvents that will melt the paint and some plastics. I found this out to my cost many years ago when trying to stick down the edges of a vinyl roof.......
As to whether to keep them on or not is up to you. As Chris said, they do offer some protection to the door edges and help break up the slab sided lines of the Uno. But then again, some people like the smooth look so it is purely down to personal taste.
Bear in mind that if you do remove the trim and tape/ glue without damaging the paint, the paintwork will probably be darker where the trim was as it has been shielded from the damaging effects of sunlight. Sometimes T-Cutting can help blend the areas in, but depending on how much your paintwork has faded it may or may not be that noticeable.
Oh, and tell your mate to be more careful when opening the doors on your car!