That’s a common misconception. The word “barba” in Spanish and “barbaroi” in Greek have distinct origin of the gibberish “bar bar bar” which is apparently how all barbarian speak. Here’s an excerpt from Wikipedia page on barbarians under “Etymology”.

The Ancient Greek name βάρβαρος (bárbaros) ‘barbarian’ was an antonym for πολίτης (politēs) ‘citizen’, from πόλις (polis) ‘city’. The earliest attested form of the word is the Mycenaean Greek 𐀞𐀞𐀫, pa-pa-ro, written in Linear B syllabic script.

The Greeks used the term barbarian for all non-Greek-speaking people, including the Egyptians, Persians, Medes and Phoenicians, emphasizing their otherness. According to Greek writers, this was because the language they spoke sounded to Greeks like gibberish represented by the sounds “bar…bar…;” the alleged root of the word bárbaros, which is an echomimetic or onomatopoeic word. In various occasions, the term was also used by Greeks, especially the Athenians, to deride other Greek tribes and states (such as Epirotes, Eleans, Boeotians and Aeolic-speakers) and also fellow Athenians in a pejorative and politically motivated manner. The term also carried a cultural dimension to its dual meaning. The verb βαρβαρίζω (barbarízō) in ancient Greek meant to behave or talk like a barbarian, or to hold with the barbarians.