The recently introduced feature of tighter integration between Bamboo and stash where when the 2 apps are linked through application link, the repositories from stash are consumable in Bamboo works in such a way that bamboo saves a key in the user profile for authentication. This feature was introduced in Bamboo 5.6. See https://confluence.atlassian.com/display/BAMBOO/Stash

Stash must have SSH access enabled: When you link to a Stash repository, Bamboo generates an SSH key pair that it shares with Stash. See Enabling SSH access to Git repositories in Stash. If you want to view the public SSH key in Stash, see SSH access keys for system use.

It will be nice to have this key shared and stored against the projects in Stash instead for the reason that the user in stash could be deactivated in future. For example if the user no longer works for the company. In situations like this, the build in Bamboo will fail and we need to login as a different user to update all the linked repositories to get this working