Yes, you can do that. You don't have to push twice but just once to push to both remote repositories.
I had the same issue before so wrote how to do it here.
Git: Push to / Pull from Both Github and Bitbucket
Of course silver lining has a cloud, and in this case, it is that while we can push to multiple URLs simultaneously, we can only fetch from the original “origin” (you can change this, but that is out of scope for this post).
Finally, to see which remote will be fetched from:
I have a similar kind of situation with a twist. Like I have an existing BitBucket branch(s) where I was managing or updating my code form past few months.
Now my requirement is to upload my project in GitHub with the only single 'master' branch.
Everything up-to-date
Branch 'master' set up to track remote branch 'master' from 'origin'.
remote: Repository not found.
fatal: repository 'https://github.com/<USERNAME>/<PROJECT_NAME>.git/' not found
Step 5 (SSH key setup) -
After few hours of investigation I have found that this is SSH key issue.
So I generate SSH key for both BitBucket & GitHub and add those keys in my both respective account.