mercurial's notion of a monotonic commit counter (that you're looking for if I understand you correctly) has about the same quality as git describe's commits-since-last-tag counter.

 

For a global numbering scheme, you could just (automatically) tag every single commit of a certain branch (master, release, ...) with a monotonic counter. That's more or less what the svn-to-git translation of tianocore is doing, for example (with tags following the scheme svn18670). That way, those numbers are actually global, unlike what mercurial is doing.

Assume that the scheme is to use $version-$number since release: git describe would emit something like 0.9.9-40-5-gabcde, meaning that your tree is based on the tag 0.9.9-40 (globally unique), is 5 local commits away from that, with your current local commit id being abcde.