Is 'Allah' the same god of the Jews and Christians?
Yes. The word in Hebrew came from "El" and in Arabic from "Elh". The word for "god" (note the small 'g' in English) is "Elah". Compare this to "Eloi" and "Eli" in the New Testament.
The word "Allah" is the perfect word to describe the God of Jews and Christians as it does not permit gender nor plural. Therefore, when the word is used as in "Eloihim" or "Allahumma" (these would appear at first to be plural and/or female genders) it becomes clear it is the royal usage such as a king would use in referring to his royal station while make a decree of some type. The king would state, "We decree the following..." and he is actually talking about his royal position rather than plural status.
There is no word in English for the "God" of Israel, the "God" of Jesus. This explains why the spelling remains the same whether speaking of a false "god" or pagan idol and the "God" of Abraham or Moses or Jesus (peace be upon them). Note the use of the capital "g" (G). This is the only way English can present a difference between the two. In the Semitic languages it is easy to distinguish the difference between a "god" and "The God" due to the structure of the word itself.
Evidence for this is quite simple. Visit any motel or hotel and remove the Bible located in the drawer next to the bed (placed they courtesy of the Gideon Society). Note inside the first few pages a reference to the translations of the Bible they have made into 27 languages. The second example they give is to the Arabic language of the passage in the New Testament from the Gospel of John 3:16. The verse begins in English; "For God so loved the world.." and in the Arabic translation the word used for "God" is "Allah."
The Arabic Bible beginning with the Book of Genesis uses the word "Allah" to represent "The God" of Creation; Adam and Eve; Noah; Abraham; Ishmael and Isaac and Israel. Page one of Genesis has the word "Allah" 17 times.