I definitely recommend Searchable Selector from Marketplace.
This is probably the best widget to handle such needs. It has great configuration options, and my favourite is that the list can have any type of content.
For mobile app, you could consult this page:
and then particular React Native Prop styles.
For web app, I advise to check HTML "select" tag and what CSS attributes you can change there. Or apply your styles on it's wrapping container. Maybe others could advise here better.
For a custom pluggable drop-down widget, I'd check this whole section: