PROBLEM SOLVED
The solution was (on Mac):
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revoke the old Access Token from GitLab (which had only the next Scopes: read_repository, write_repository)
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generate new Access Token having the next Scopes: api, read_user , read_repository, write_repository (it was strange that for other colleague it worked only with read_repository, write_repository)
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on SourceTree: go to SourceTree → Preferences ->Accounts → Add → fill-in the details as in the following image (and for Password use the Access Token)
after this, I was able to Clone Repository (“Clone from URL”)
also, this link inspired me: https://www.awordfromnet.com/access-gitlab-via-sourcetree-updated/