Greek writers who came east with and after Alexander described an Indian god they called Dionysos, and ancient texts of that period call Shiva the Indian Dionysos and Dionysos a god of the Orient. The grounds offered for the identification were the bull, the snakes, the dancing, the ecstatic following and the freedom from ordinary restraint. This is interpretatio graeca: the habitual Greek practice of naming a foreign god by the nearest Greek one rather than learning a new name, and it is evidence about Greek habits of mind before it is evidence about Shiva.
Two distinct claims get confused here and should be kept apart.
The first is that the ancient identification was made. That is well attested and is a fact about the Hellenistic period. Megasthenes, whose Indica survives only through later writers, principally Arrian, gave the Indians a Dionysos who founded cities, gave them laws, taught them to sow seed, and stood at the head of a list of a hundred and fifty-three kings running down to Chandragupta. That is a civilisational founding-myth in Greek dress, and it is doing chronological work as much as religious work.
The second claim is that the identification reflects a real historical relationship, whether a shared Proto-Indo-European inheritance or a later borrowing. Roger Woodward has noted parallel phallic symbolism across Indo-European religions and suggested a connection; others argue for indigenous and tribal origins for Shiva that owe nothing to Europe. That argument is open.
The same care applies to the Herakles the Greeks reported. Arrian places him among the Surasenians, in a territory containing Methora and Cleisobora on the river Iobares. Mathura on the Yamuna is the Krishna heartland, and Edwin Bryant has argued on that geography that the Greek Herakles is Krishna-Vasudeva. Others have proposed Balarama or Indra. The reading remains debated, and the atlas attributes it rather than asserting it.