I have a lot of files with different coordinate systems. However, coordinate extractor only works to get a coordinate system of a file. Is there a way to get a list of coordinate system names of multiple files in different formats (shp, tab, kmz..)

I think you can achieve this with a Directory & File Pathnames Reader and a FeatureReader. The Directory Reader can be set up to read in all the files you want to check. And then run that into the FeatureReader setup with Format: Generic (Any Format) and Dataset: path_windows attribute from the Directory Reader. Then run the Generic port into the CoordinateSystemExtractor. Keep in mind you will have to clean up the result, especially for multi-file formats like shapefiles. This method reads in all the files individually (.dbf, .shp, .shx, .prj, etc.) of which only the .prj file will have a coordinate system assigned to it.