There is a problem with the Hebrew text which compounds the "trickiness" of this text in 2 Kings 2:14.
Benson's commentary is the most helpful here.
2 Kings 2:14. And said, Where is the Lord God of Elijah? — Who at
Elijah’s request divided these waters, and is able to do it again. But
according to this translation, two words are left out, namely, אŠ הוא,
aph-hu. The clause literally rendered is, Where is the Lord God of
Elijah, even He? which a learned foreigner interprets thus; that
Elisha having asked this question, Where is? &c., answers himself in
the two last words, aph-hu, yea, he is yet in being. Abarbinel
expounds them, Though Elijah be not here, yet his God is. The servant
is wanting, but not the Lord. The blessed God is still present, and
will supply his place. And when he also had smitten the water’s, they
parted hither and thither — As when Elijah smote them with the same
mantle, which they both used, as Moses did his rod, not imagining that
there was any inherent virtue in it, or at all trusting therein; but
using it as a mere sign of the presence and power of God, in which
alone they confided to work this wonder. Thus Elijah’s last miracle
was Elisha’s first, and the disciple began where his master left off,
taking up and carrying on the same blessed work of witnessing for God
against idols and idolaters.
That is, putting the textual problem aside, Elisha is asking, "Will God be with me as He was with Elijah?"
It was the act of catching Elijah's mantle that has now passed into the language, meaning, the one who now wears the mantle is the recognized successor. God then demonstrated this by performing precisely the same miracle as Elisha had seen Elijah do and with the same mantle. Elisha had now taken Elijah's place as the recognized prophet to Israel.