What I find often is that if for example I get 40 shapefiles from County X, they will all be in the same County X projection even if some of them are missing the PRJ files. So it's a good bet that you can force the data into the same projection by copying the PRJ file.
For example, if you have a shapefile called "okaydata" and one called "funnypoints" then you would copy "okaydata.prj" to "funnypoints.prj" and it would then be defined with the same projection.
Of course there is an impressively complicated tool with colorful buttons you can use to do all this in ArcGIS Pro but sometimes copying a file is easier. I think it is called "Define Projection" The tool's advantage is that it works the same way for file geodatabases as it does for shapefiles.