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TL;DR: Yes.
A minecraft server processes and stores personal data, at least player UUIDs and IP addresses. Depending of what plugins a minecraft server has enabled, more personal data could be affected. In fact, almost none of the existing minecraft servers / server owners care about data privacy.
No. If a minecraft server is publicly available, every server owner is responsible for respecting data privacy regulations of all players, because a player could also reside inside the European Union.
There is a good [description][https://gdpr.eu/eu-gdpr-personal-data/] of what personal data is considered; however, every data that can be related or attributed to a natural person is considered being personal data.
This page is meant to be read by server owners and plugin developers/maintainers. If you are not belonging to either one of them, none of this might be of interest for you.