How we can reproduce it using the example you attached
1 Create new project with unique (the project with such name does not exist in your ORG on Services web site) name - when Editor opened - notice that Services are enabled in Connect window - please see image "Step1" in comments
2 Close the Editor and delete this project from your computer (from hard drive) - this is just to be able to create project with the same name in the next step
3 Create new project with the name you used in step 1 (already existent on Services web site) - when Editor opened - notice that Services are NOT enabled in Connect window - please see image "Step2" in comments

Expected result: I would expect that services Enabled and project name on services web site is "New Unity Project 6(1)"