With an advanced license, you can use
PolygonToLine_management (in_features, out_feature_class, "IDENTIFY_NEIGHBORS")
That gives you a line attribute table with FIDs for polygons on the left and right sides.
Relationships can get complicated with nested polys and polys with holes,
but essentially any line with positive LEFT_FID and positive RIGHT_FID divides the two polygons with these FIDs.
Thus, it shows ajacentcy.
Joining this table back to the polygons tells each poly its neighbor.