Open MAT files by the Matlab:
FileData = load('File.mat')
Or use the https://de.mathworks.com/help/matlab/ref/matfile.html matfile command.
I never needed to open a MAT file by a double click, but if MS Office was installed after Matlab, the .mat extensions is associated to MS Access. There are several methods to associate it with Matlab again, simply ask your favorite internet search engine for these terms.