Refactor DropdownContentsEditView component to use GraphQL + Apollo

What does this MR do?

This MR is a part of &6092 (closed) where we refactor Labels Select component to use Apollo client + GraphQL instead of REST + Vuex. In this particular MR, the changes are mostly applied to dropdown_contents_labels_view.vue.vue component.

To test the changes locally, you will need a feature flag enabled:

Feature.enable(:labels_widget)

After enabling FF, you can test issue sidebar labels.

Screenshots (strongly suggested)

No visual changes

Does this MR meet the acceptance criteria?

Conformity

• I have included changelog trailers, or none are needed. (Does this MR need a changelog?)
• I have added/updated documentation, or it's not needed. (Is documentation required?)
• I have properly separated EE content from FOSS, or this MR is FOSS only. (Where should EE code go?)
• I have added information for database reviewers in the MR description, or it's not needed. (Does this MR have database related changes?)
• I have self-reviewed this MR per code review guidelines.
• This MR does not harm performance, or I have asked a reviewer to help assess the performance impact. (Merge request performance guidelines)
• I have followed the style guides.
• This change is backwards compatible across updates, or this does not apply.

Availability and Testing

• I have added/updated tests following the Testing Guide, or it's not needed. (Consider all test levels. See the Test Planning Process.)
• I have tested this MR in all supported browsers, or it's not needed.
• I have informed the Infrastructure department of a default or new setting change per definition of done, or it's not needed.

Related to #332257 (closed)