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InfluxDB
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The custom element spec definitely only deals with the mechanics of when are where to run your component's lifecycle code - it says nothing about data. So your choices are basically property accessors, which are interoperable, but require prop-drilling for global-ish data, or something proprietary like a state management library.
The Web Components Community Group (WCCG) is offering something of a third way with the community protocols: https://github.com/webcomponents-cg/community-protocols
The Context protocol provides tree-scoped ambient data in an interoperable way. It's implemented by Lit and FAST (I believe). It doesn't replace a data store, but it's often used to provide data stores to components, and at least reduce some coupling.