There was confusion in the roles, for sure. That is the source of the conflict and drama in
Genesis 16.
As the commentaries have stressed, Hagar almost certainly had a lesser role than Sarai. The purpose of having a child for Abraham in Sarai's stead implies that the status as wife/concubine was temporary.
However, Hagar took advantage and thought she could take the priority in Abraham's heart because of the son. We don't know how tender the relationship between Abraham and Hagar had become. We do know Hagar began to "despise" Sarah (
Genesis 16:5). She wanted her new status with Abraham to break her bondage to Sarai.
Abraham was wise in his response. He clarified lines of authority. Hagar was totally under the authority of Sarai, and Abraham claimed no right to interfere: "Your servant is in your hands. Do with her whatever you think best" (
Genesis 16:6).
Later God confirmed this status for Hagar, telling her to return and submit to Sarai (
Genesis 16:7-9).
So, to the question, let us use the context (which is all we have) to bring clarity to the meaning of the word "wife" in
Genesis 16:3. It seems that Sarai was magnanimous in giving Hagar to Abraham, and may have even considered it more than just a concubine, not having the wisdom to see the potential problem.
Hagar wanted to take advantage and have her position become that of a true wife, on equal status as a competitor to Sarai. This suggests that this was a situation in flux and there probably was not a clear legal unbending status that would fit our categories of wife or concubine. It was the relationship that would determine the status, not the legal status that would determine the relationship.
Thus, Abraham's response was most decisive. He determined clearly and unequivocally that Hagar was no more than Sarai's servant, having stolen none of Abraham's heart, and who could not call on any relationship with Abraham to save her from Sarai's wrath. By Abraham's decision, Hagar became nothing more than the bearer of Sarai's child.
So, it seems best to say that, regardless of Sarai's intent in
Genesis 16:3, Hagar did not have the status of a wife of Abraham