It is extremely common to capitalize such terms in rules, commentary and studies of games. Any reference to "a player" is usually left un-capitalized but a reference to "Player 1" typically is.
1) Let us assume the first player is 1.
2) Let us assume Player 1 moves first.
3) Whenever a player moves [...]
A common exception to this rule is when the players will be referred to without the word "player":
4) Let us assume player 1 moves first. If 1 moves by doing [...]
This is more apparent when "A, B, C" are used instead of "1, 2, 3":
5) Let us assume player A moves first. If A moves by doing [...]
However, none of the above is encoded in a grammatical rule. My impressions of what is common comes from having read a great many manuals and instructions for games of all sorts.
If you are concerned about your own publication, then I recommend checking with your publication's style guides or with your editor. If you are merely concerned about someone else's usage, then they apparently disagreed with your personal style. There isn't a hard and fast rule on this subject.
... which isn't to say the subject is not worth studying from a user experience standpoint. I assume the advantage in capitalizing the term is to allow for better skimming. This could easily explain why (4, 5) do not capitalize while (2) does: The capitalization is telling you what to look for in the future.