About the real projective plane, it is a space in which -- each "point" is a pair of opposite points on a Euclidean sphere -- each "line" is the set of "points" on a great circle of that sphere. So, on the projective plane, any two lines intersect in /one/ point.
A finite projective plane can be constructed having qq + q + 1 points, and the same number of lines, where q may be any prime power. The number of points on a line, and the number of lines through a point, turns out to be q+1. E.g. when q=2 there is a projective plane with 7 points 0,1,2,3,4,5,6 and the seven lines 0 1 4 0 2 5 0 3 6 1 2 3 1 5 6 2 4 6 3 4 5 You can verify that any two points are on one common line, and any two lines have one common point. http://en.wikipedia.org/wiki/Projective_plane
Back to the game, I wouldn't get too cosmic about the Mobius strip. Here: a b c g h i ... d e f d e f ... g h i a b c ... The figure represents a matrix of 9 points on a Mobius strip. Each point of the Mobius strip is represented by a /pair/ of points on a torus. If you follow the row abcgh... to the right, it wraps back into the 3*3 square on the left in the desired way. When a player makes a move, he takes /both/ c's or /both/ e's or whatever. Some construction of this sort might yield an interesting game. I wouldn't worry about how to draw the /grid/ as we do in tic-tac-toe; it is the cells of the grid, and the list of sets of cells which comprise winning "rows", that matter.