Migrate deprecated neutral classes to design tokens in FilteredSearchTerm

During MR !5113 (merged), it was identified that FilteredSearchTerm still uses old neutral utility classes (e.g., gl-bg-gray-10, gl-text-gray-900) directly instead of design tokens.

We should replace any remaining deprecated utility classes with design token–based CSS variables, consistent with the changes made for view-only mode. Use appropriate control-* tokens wherever possible to match the design intent (e.g., background, text color).

See notes in !5113 (merged) (comment 2461110643) for details:

• src/components/base/filtered_search/filtered_search_term.vue#L230

• src/components/base/filtered_search/filtered_search_term.stories.js#L34