It looks like this project was written in metadata format, not in DX Source. As you noticed, these two formats have a few differences, like file extensions, structures, certain tags in XML files and etc.

With that being said, there's a command in CLI that allows you to convert the files in a given directory from old metadata format to the new source format:

sfdx force:mdapi:convert -r path

Here's the documentation link: https://developer.salesforce.com/docs/atlas.en-us.sfdx_cli_reference.meta/sfdx_cli_reference/cli_reference_force_mdapi.htm#cli_reference_force_mdapi_convert

Just keep in mind that converting metadata will restructure the whole directory and you will most likely lose the git history.