The claim
"'Allah' is a different deity from the God of Judaism and Christianity."
The short answer
'Allah' is simply the Arabic word for 'God.' Arab Christians and Jews have used the word 'Allah' for centuries. Islam affirms the same God worshipped by Abraham, Moses, and Jesus, and considers itself a continuation of the Abrahamic prophetic tradition.
Evidence-Based Clarification
Allah Is the Arabic Word for God
The word 'Allah' predates Islam and is used by Arabic-speaking Christians and Jews to this day. Arabic Bibles use 'Allah' for God. The Quran explicitly states that Muslims worship the same God as the People of the Book (Jews and Christians) and affirms the prophets of the Hebrew Bible and the Gospel.
PRIMARY SOURCES (2)
"And do not argue with the People of the Scripture except in a way that is best... and say, 'We believe in that which has been revealed to us and revealed to you. And our God and your God is one; and we are Muslims [in submission] to Him.'"
— Surah Al-Ankabut 29:46check the original
"Say, 'We have believed in Allah and what has been revealed to us and what has been revealed to Abraham and Ishmael and Isaac and Jacob and the Descendants and what was given to Moses and Jesus and what was given to the prophets from their Lord. We make no distinction between any of them, and we are Muslims [in submission] to Him.'"
— Surah Al-Baqarah 2:136check the original