`base-controller`: Replace use of `any` with proper types (non-test files only)

[mcmire]

Look for the following string in non-test files to find these: // TODO: Replace `any` with type

These are occurring in BaseControllerV1 - so we are deciding to not address this.

[MajorLift]

Both instances are examples where as any is used to add new properties to an object at runtime. This is a use case that will be listed as acceptable in the TypeScript guidelines any section.

We could replace the TODOs with comments that explain this fact and point to contributor-docs. This would serve as a good example case for 2024/Q1/O2/KR2.

If we go ahead with this approach, we should add MetaMask/contributor-docs#47 as a blocker for this ticket.