Abba is Aramaic not Arabic. Aramaic was a major language as early as the 10th century BC, while Arabic did not become a major language until after the 7th century AD. Both languages are part of the Afro-Asian language group. Therefore, when Muslims appeal to similarities between Hebrew/Aramaic words they are appealing to the same language group.
The Arabic word translated Allah and the Aramaic word translated God are derived from the same Semitic language group word, therefore based on the history of languages and the etymology of both Aramaic and Arabic Muslims are appealing to myths when they assert that Jesus spoke Arabic or that Arabic was even known by the vast majority of 1st century Palestinian Jews. On the other hand since Palestine was a Roman province and Koine was the language of government and commerce in the Roman Empire it is not unreasonable to believe that Jesus could and did speak Koine when necessary.
The God (Allah) that Jesus knew (knows) is not the god of Islam, since the God of the OT required the Jews to treat aliens in their communities with the same respect they treated other Jews.
Deuteronomy 10:17-19
17 For the LORD your God is the God of gods and the Lord of lords, the great, the mighty, and the awesome God who does not show partiality, nor take a bribe. 18 He executes justice for the orphan and the widow, and shows His love for the alien by giving him food and clothing. 19 So show your love for the alien, for you were aliens in the land of Egypt.
NAS
The God of the OT and the NT tells us "vengeance is mine, I will repay". The god of Islam tells Muslims to slaughter unbelievers.