The Kanban board has quick filters at the top the user can click on and filter the data based on those, but if the user wants to filter just for themselves, can they do a basic search or write some jql to filter it themselves, just temporarily? I don't see this option on the kanban screen, only when searching for issues on the issues tab.
No, but yes.
The board is a shared object, set up by the board owner and administrated by them. That includes the quick-filters, and non-admins cannot change that (the point of a board is that the users all see the same thing and work on the same information).
But there's nothing to stop them using the board filter as a basis for their own filtering, or, if you let them, create their own board that looks at the same filter, but has their own quick filters on it.
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