The cloak was the otter clothing of a prophet. It was the sign of his job. When Elisha took Elijah's cloak he was saying he was taking Elijah's place as prophet.
Elisha is asking if Yahweh is with him as he was with Elijah. AT: "Yahweh, the God of Elijah, are you here with me?" (See: [[rc://en/ta/man/translate/figs-explicit]])