Avestan haoma and Vedic soma are the same word. Both come from an Indo-Iranian form built on a root meaning to press or to extract, so the name describes what is done to the plant rather than what the plant is, which is a large part of why identifying it has proved so hard: the ancestors of both traditions named the substance after the act.
The correspondence is not confined to the noun. Both traditions press a plant for juice, filter it, mix it with milk or water, and both make the resulting drink the centre of their principal sacrifice. Both personify it as a divinity who is also the offering. Both attach immortality to it. Avestan and Vedic are close enough that whole lines can be transposed between the Gathas and the Rigveda by regular sound correspondence, and the haoma and soma passages are among the clearest cases. The reasonable conclusion, and the one most Indo-Iranianists draw, is that the ritual is inherited rather than borrowed: there was a pressing rite before there was an Iran or an India.
That makes the identification question a single question with two sets of evidence. Whatever the Indo-Iranian plant was, it should account for both bodies of description, and any candidate that explains the Vedic material but not the Avestan is doing only half the work. It also means the living Zoroastrian use of ephedra is the strongest single piece of evidence anyone has, because it is an unbroken practice rather than a reconstruction, which is why ephedra is taken seriously for soma by scholars who find nothing in the Rigveda itself to suggest it.
What continuity cannot establish is that the plant never changed. Substitution is documented on the Indian side, and nothing rules it out on the Iranian side.