If you want to create create multipoint Feature Class, here is what you could do.
2. In the above Polygon Layer's Attribute Table, add a New Field (eg. Name: Zone_ID. Datatype: Text). Fill this field with unique values. You could also fill it with the values of FID.
3. Use Spatial Join to join the attributes of the Polygon Layer to the Points Layer. (Target_Feat: Point Layer, Join_Feat: Polygon Layer, Join_Type: ONE_TO_ONE, Match_Option: Intersect)
4. In the output Point Layer of the above step, run Dissolve tool (Dissolve_Field: Zone_ID). Keep Create multipart features Option Checked.
The output of Step. 4 will be a multipoint Feature Class, where each record in attribute table will reflect multiple points on map.
Let me know if there is any concern.