Thanks to everyone for voting and commenting on this suggestion. In Bitbucket Server and Data Center 5.5, we have introduced Personal access tokens. User can create a token with limited scopes (e.g project and repository read) and use them for authenticating REST calls, git operations and integrating with other applications. More details can be found in the release notes.
This satisfies the original suggestion but some of you have requested the the ability to add OAuth 2.0 consumers for integrating with other applications. It is something that we're exploring and I'm keeping this issue open to track it however, we're not able to provide any timeline for it. Learn more about our process here.
Product Manager - Bitbucket Server and Data Center