RCS keyword substitution is not natively supported by Git, but can be added with a gitattributes
filter driver: See "Git equivalent of subversion's $URL$ keyword expansion".

For example (not exactly relate to your question, but illustrates the general principle):
git config filter.rcs-keyword.clean 'perl -pe "s/\\\$Date[^\\\$]*\\\$/\\\$Date\\\$/"'
git config filter.rcs-keyword.smudge 'perl -pe "s/\\\$Date[^\\\$]*\\\$/\\\$Date: `date`\\\$/"'
You will base your filter script on the result of git describe --tags
called from your Makefile.
As mentioned in this answer to "Git equivalent of subversion's $URL$ keyword expansion", smudge/clear filter driver is not a perfect solution, and adding any kind of meta-data directly in the data (source) is generally a bad idea (you have a debate about it in "What are the basic clearcase concepts every developer should know?").
Yet you have a good example of such Git keyword expansion in this answer in "How do I enable ident string for Git repos?".