HTML5 <video> can play .mkv files?

HTML5 doesn't support any video formats, or rather HTML5 doesn't specify what formats browsers should support. It's up to the browsers to decide which formats they choose to support. Apparently Chrome plays .mkv, but I wouldn't be surprised if other browsers didn't play the same file.

The website has probably set the mimetype of the file to video/webm. Chrome will open this inline rather than forcing a download (because otherwise it doesnt know what to do with it or how to open it).

E.g., in apache this is done by using the AddType directive (can be done in a htaccess):

    AddType video/webm .mkv