feat(dropdown): Persist filter dropdown value between pages

[ethan-id]

Description

Adds a composable useApiFilter() to persist the user’s selected filter across page navigations and reloads. Wires it into FilterDropdown.vue and the categories page so the persisted value is used on initial load and when switching categories.

Proposed Changes

• New frontend/composables/useApiFilter.ts: stores the current filter in useState('apiFilter') and syncs to localStorage on the client.
• Update frontend/components/FilterDropdown.vue:
  • Bind v-model to the shared filter from the composable.
  • Emit the current value on mount so parent views initialize correctly.
• Update frontend/pages/categories/[category].vue:
  • Use the shared filter and refetch when route.params.category changes (watch(..., { immediate: true })).
  • Remove redundant local filter state and the extra onMounted fetch.

Checklist

• My submission is formatted according to the guidelines in the contributing guide
• My submission has a useful description
• Code follows the established CODE_OF_CONDUCT guidelines.
• Code has been tested thoroughly.
• All commit messages are descriptive and follow the established commit message format.
• Pull request description clearly describes the changes included in the pull request.
• I have searched the repository for any relevant issues or pull requests
• Any category I am creating has the minimum requirement of 3 items

[imggion]

Thanks for your contribution. We'll review this pr soon! 😄