[json-rpc-middleware-stream] fix: clean up engine listener

[MajorLift]

Port the following PR that was ongoing in the @metamask/json-rpc-middleware-stream repo prior to its migration into the core monorepo.

• fix: clean up engine listener json-rpc-middleware-stream#23 (author: @iantanwx)

There are some conflicts to resolve due to test rewrites that happen here: MetaMask/json-rpc-middleware-stream#25.