The easiest connection from Oesho to Vayu, also known as Vayu-Vata, is actually through their names. It's not mentioned in the wiki article you linked, but there's an alternate spelling on some Kushan coins where it is spelled Oedo (ΟηΔο) rather than Oesho (Οηϸo). The connection between Vayu-Vata and Oesho may not be obvious, but there is a general pattern of contracting and compressing words in Iranian linguistic history. Ancient Avestan Vayu-Vata may have passed down to Middle Bactrian and changed significantly

This etymology is only theoretical because both earlier Bactrian and late Avestan are completely undocumented. However, based on patterns seen in languages like Persian and Parthian it may have gone something like this:

• Drop the V and it becomes Ayuata

• The long string of vowels "Ayua..." blended together in normal speech. "Y" is a relatively subtle sound and probably vanished altogether. - Auata

• The initial syllable got rounder to stand out passing through something like "Ah" before becoming an "O" sound. - Ouata

• "ua" is a kind of difficult pairing to make it many languages, especially following an O, and likely blended into the "ay" sound as in "hay" represented by the letter eta (η) in antiquity. - Oeta

• The shift between T and D either is relatively common across all languages. - Oeda

• Why exactly D would become Sh isn't clear to me, but both appear on Kushan coins, so it clearly happened. - Oesha.

he very obviously looks like Shiva to me, but some scholars claim that he is the Indo-Iranian god Vayu.

These are not necessarily conflicting statements. As the Wikipedia article you linked notes, the similarities to Shiva are well documented. Indian images of Shiva share a lot of iconography with Kushan images of Oesho. It's harder to talk about any specifics in theology or mythology because we don't have a lot of Kushan sources besides coinage.

In the Rig Veda, the oldest extant evidence for Indo-Iranian religion, shiva/śiva is not the name of a specific god. Instead, it's just one of several titles associated with Rudra. His other titles include vayu and vata, meaning "of the winds" and "of the air" respectively. In Iranian religion Vayu, or Vayu-Vata, was a god's name. In the surviving sources, Vayu is relatively obscure, but some or the blessings associated with Vayu in the Ram Yasht are similar to some ideas associated with Shiva in India too.

Vayu is also associated with Rama in Zoroastrianism, another deity shared with the Vedas and Hinduism. Of course, Rama occupies a similar level of supremacy to Shiva in some strains of Hinduism, either independently or as an avatar of Vishnu. So there's another layer of beliefs that might lead someone coming into the Indo-Iranian sphere to equate Vayu, Rama, and Shiva together as similar or the same.