Using 2 code repos, one Acquia and one private gitlab server for Drupal code base tracking I run a few drupal 7 sites on acquia and I also have a separate gitlab server for sites that are not on Acquia. Is it possible to push to both repositories on Acquia and Gitlab?
Definitely possible and it's what I do. Acquia provides some documentation  on this. As Kevin mentioned you can do this by setting up two remotes for your local repository clone, one at GitHub/GitLab/Bitbucket, and one at Acquia Cloud.