I agree with @BERA's solution if you are trying to count the number of points in a polygon, and it is safer in many cases than relying on pre-populated point feature attributes. But if you simply have a table of polygon IDs and you are wanting to ignore the polygon geometry and just get the count of the number of times a specific polygon ID shows up in the point feature class's attribute table, then a table summary may work for you.
In ArcMap, if you open the attribute table of the point FC, right click on the FID_2 field and select "Summarize...". The output of that operation should be a table of the FID_2 values and the desired count field.
Then you can bring that newly created table into ArcMap and do a table to table join (http://desktop.arcgis.com/en/arcmap/latest/manage-data/tables/essentials-of-joining-tables.htm), joining the summary table to the point FC based on the FID_2 field.
Finally you can run Calculate Field on the Point FC's Count field, populating it with the value of the summary table's count cnt_fid_2 field (that field name may vary slightly).