Hi there,
There are currently no default Remove Duplicate Label options in ArcGIS Pro when working with point layers. However, here is a temporary solution that may work for you:
1. Create a polygon buffer around the point layer with a small distance value, like 1 inch.
2. Configure labeling of this new buffer layer as you would for the points layer. Ensure the Maplex label engine is enabled ("More" dropdown in the Labeling tab > Use Maplex Label Engine). The option to remove duplicate labels should be available now for the buffer layer.
3. Set the fill and outline color of the buffer layer to "no color."
Hope this works for you!
- Rachel