[Remove Vuetify from Studio] Create public API and fine-tune autocomplete multi-select component

[MisRob]

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Description

Now when https://github.com/learningequality/studio/issues/5133 is set up, continue in https://github.com/learningequality/kolibri-design-system/pull/1087 by adding public API suitable for all our use-cases, and also fine-tune the component.