I think this is a tool which might help you.

PolygonNeighbors_analysis

http://pro.arcgis.com/en/pro-app/tool-reference/analysis/polygon-neighbors.htm

I think this is a tool which might help you.

I tested it with sample data within ArcMap 10.5 using links below.

PolygonNeighbors_analysis

http://pro.arcgis.com/en/pro-app/tool-reference/analysis/polygon-neighbors.htm

How it works:

http://pro.arcgis.com/en/pro-app/tool-reference/analysis/how-polygonneighbors-analysis-works.htm

Output is a table which will have two main coulumns src_OBJECTID (with id of polygon) and nbr_OBJECTID (ids of polygons near).