ANSI color in git is not displayed correctly

Try setting your LESS environment variable to include the -R option:

LESS=-R git grep ...

If this works, add export LESS=-R to your ~/.bashrc or ~/.profile or similar shell startup file.

   -R or --RAW-CONTROL-CHARS
          Like -r, but only ANSI "color" escape sequences are
          output in "raw" form.  Unlike -r, the screen
          appearance is maintained correctly in most cases.
          ANSI "color" escape sequences are sequences of the
          form:

               ESC [ ... m

          where the "..." is zero or more color specification
          characters

The problem, as others have noted, is that your terminal is fine, but when Git invokes the pager, it's not interpreting the ANSI color codes correctly.

I'd start out by unsetting LESS in your environment; it sounds like you might have previously been setting it to something obscuring what Git needs. If that solves it, there you are. If you really must customize LESS, note that Git starts out with FRSX as default, so be wary of changing those if you don't need to.

If you do for whatever reason want LESS in your environment different from what you want for Git, the ideal way to deal with Git and the pager is through the core.pager config variable. To quote the manpage:

The command that git will use to paginate output. Can be overridden with the GIT_PAGER environment variable. Note that git sets the LESS environment variable to FRSX if it is unset when it runs the pager. One can change these settings by setting the LESS variable to some other value. Alternately, these settings can be overridden on a project or global basis by setting the core.pager option. Setting core.pager has no affect on the LESS environment variable behaviour above, so if you want to override git’s default settings this way, you need to be explicit. For example, to disable the S option in a backward compatible manner, set core.pager to less -+$LESS -FRX. This will be passed to the shell by git, which will translate the final command to LESS=FRSX less -+FRSX -FRX.

That, combined with some knowledge of the options you want, should get you where you want to be. (The fancy backward-compatible method works by disabling all options currently in LESS, then adding back in the ones you want.)

In git, you can change your pager to use the -R option:

git config --global core.pager "less -R"