I had the same issue when I wanted to copy/paste point features from two different feature classes (parcPoint_5 and parcPoint_6) to one feature class (parcPoint5_6). I added all three to a Pro project map through Add Data tool.
First, I left click one feature class and selected all features and in Edit menu copied to clipboard. Then I found ‘Past’ dropdown arrow is grayed out, no ‘Paste Special’ tool. When I pasted it simply add a same layer again to the map. (I think Pro is designed this way to protect data.)
Second, I selected all features with the Select arrow of the edit Manu, and copied to clipboard. I found dropdown arrow activated. However when I try to past it to the new layer, with Past Special tool, error window popped up ‘This data is currently locked by another user, cannot be saved.
Then through Pro database connection I add the database to the catalog and moved the data set with three feature classes to the ArcPro project’s default database. Removed the three feature classes from the map and drag and dropped three feature classes from the project’s default database connection. Since then copy paste all worked fine in the Edit menu tools.
Hope this helps!