Trademarks exist to avoid confusion in the public about who or what a particular name refers to. Trademark right are not given automatically, but trademarks have to be explicitly claimed and actively defended.
A quick search shows no trademark registration on the name "Twine" in the context of Python packaging, nor any statement in the project's repository and documentation that Twine is used as a trademark. There is a Twine trademark for a tool for "telling interactive, nonlinear stories".
As trademarks are intended to prevent competitors to use "confusingly similar" names, the scope of a trademark's protection is in part determined by how the name in question is known among the general public. The name "Twine" is not associated in the public's mind with one particular company, so the trademark's protection is effectively limited to the market where the holder of the trademark operates.
If there would be a trademark on PyPA's Twine, then the question becomes if your name "twine-exists" would be interpreted (by the average Python user) as a third-party add-on to PyPA's Twine or as something offered by PyPA as a tool belonging to or replacing Twine. For the first case, the use of the trademark's name in your name is typically allowed (but it is good to get explicit permission), and in the second it isn't. Personally, I would read "twine-exists" as a PyPA tool to check if Twine is (properly) installed.