No.
Ilah is the Arabic word for G-d. Allah is the name of the moon rock that the Muslims worship.
Any Christian or Jew that calls G-d, "Allah" is being politically correct and appeasing to the Muslims.
Not quite so fast. The first time I attended a Turkish-language Mass in Istanbul, it was a bit of a shock to hear the Creed begin with belief in Allah. However, that is simply the Turkish word for God. It doesn't mean that Turkish Catholics believe in the Islamic Allah.
As for whether the Islamic Allah is the same as the God of Christians and Jews, one has to look at the characteristics of God and Allah, as described by their respective followers. As pointed out earlier, the Christian God is Triune; the Islamic Allah is not. The Christian God is a God of love; the Islamic Allah is a god of vengeance. The Christian God warns that murderers will go to Hell; the Islamic Allah promises that murderers of infidels will go to a heaven populated with dark-eyed virgins.
No way the two are the same.
It's worth remembering that "God" is not God's name. It is a description, not a name. We name things so we can tell similar items apart. God is unique; a name is not necessary. When Moses asked "who shall I say sent me?" he was expecting a name. All the gods he knew about had names. The reply was "I am." That, too, was a description, not a name. But it made the point.
I'm sorry, but I think this is a somewhat more complicated semantic tangle. The term "Allah" is commonly used by Arabic and Aramaic-speaking people to refer to the God of Abraham, the Creator and Judge of the world. Maronite, Nestorian and (I believe) Coptic Christians have been using the term "Allah" for probably over 1000 years.