The naming idea is a huge stretch. God named them both adham (Gen. 5:1-2) God named their gender, male and female. When God presented her to the man it is more likely that God said this is from-adham. And the man did not actually give her something we can consider a real name until after they were expelled from the Garden.
a. Ish and ishah are nouns, not names. Adham was simply recognizing in poetic form that the woman was from himself, a fact God had to reveal to him first.
b. When an official act of naming occurs, there is a distinctive formula. It is employed in Gen. 5:2 where it is shown that God named the two of them Adham. It is found in 2:19-20 where the man names the animals. It appears in 3:20 where the man renames the woman Eve.
c. God never instructed the man to name Eve.
d. Since God already named them both, no naming was necessary. We could think of them as the Adams, or Mr. and Mrs. Adam. That was their name, given to them by God, their creator.
e. The fact that the man renamed her Eve after they were expelled from the Garden, confirms that the man did not consider ishah a formal name, which it wasn’t.
Does the fact that the man named the animals, the means to which he was granted or given dominion over the animals? It cannot. God assigned stewardship dominion of all the creatures of earth to BOTH the man and woman before He even created them. Thus, naming anything is not a means to obtain rulership of any sort.