This happened several years ago. I joined a thrown-together 4 ball (me, another single and a 2 ball). On a hole with OB along the right side I hit a tee shot that was along the right side and it was not clear whether it was OB or fine. So I hit a provisional. The other single hit a shot that was clearly OB and he also hit a provisional. I thought his provisional ball hit the large tree that was in the center of the fairway and assumed he saw the same thing.
As it turned out, my Titleist marked with a red blotch on the "T's" was fine so I walked over to the middle of the fairway to retrieve my provisional. It was a Maxfli similarly marked but with a red blotch on the "M's". The other single objected and said that the ball was his provisional. I suggested that he walk back a bit since I was pretty sure he had hit the tree. No, he insisted, that Maxfli was his. I showed him that every ball in my bag was marked in a similar fashion and pointed out that my ball had clearly gone over the tree and there was no other ball in the area. Nope, he insisted, that was his Maxfli. I said it was a bit of a coincidence that he happened to have a ball marked exactly like mine. He said that he had found the ball earlier and that it had been marked already. He was not willing to go back and look near the tree.
We were not playing a match and this was not a tournament so I just gave him the ball but he remained angry for the rest of the round.
I am not sure what I would have done in a tournament setting. I know we often show our markings prior to the start of a round in tournament play but there are times that step is skipped or forgotten. After 50+ years of playing I never have had someone I was playing with have the exact same marking that I use (a big red blotch that covers the entire "T" on both sides).
Any suggestions as to what to do when two players claim the same ball?