Scatter chart grouping and data label

Hi Steellaa 
Unfortunately, we don't have the functionality of showing category labels in a scatter plot.

they are on the legend.
As an alternative, you can try using line graphs without lines, only markers, and customized labels.
Something like this :

pbix is attached

 If this post helps, then please consider Accepting it as the solution to help the other members find it more quickly

Yep, if you need the size too, my solutions will not help.
It seems that you need to stay with a legend...