Since I updated to macOS Mojave, DVD Player application stopped playing my old DVD files.

Again, have you tried VLC? VLC provides rather more than the Apple tools, and Apple has generally been trending away from DVD players and related tools.

If you’d like to provide Apple with feedback around this problem, contact the Apple Support folks directly, or use... Product Feedback - Apple.

The folks at Apple don’t necessarily see bug reports or enhancement requests posted here.

Yeah, I posed feedback already, and yes, I use VLC extensively, but for DVD's, it doesn't save bookmarks effectively.

Agreed, and same with me. It’d be fantastic if they could include support for dvdmedia files. And also the old bookmark system that I use a ton for teaching. Just discovered today before class it no longer works properly.

Joshua Yumibe wrote:

Yeah, I posed feedback already, and yes, I use VLC extensively, but for DVD's, it doesn't save bookmarks effectively.

Log a bug or an enhancement request against VLC too, as I’d wager that DVD and Blu-ray tool and support aren’t going to be a priority at Apple. But I might well be wrong, there.

Load and try VLC?

It plays back the files, but doesn't read the bookmarks, which is the main issue.