The 'Allah' debate has been on before I was born and till now, well-meaning Christians, theologians and missiologists have been battling to clear the air on matters like: Is Allah God? Do we serve the same God with Muslims? 
To answer properly, giving a Yes or No may not do justice to the question. History is a tool that helps in matters like this.
A BRIEF HISTORY
Allah is a word which means 'the god' and it is from the Aramaic word 'al-ilah'. It is a composite name for the highest god of the Arabs especially in Mecca. You must note that Pre-Islamic Arabs were purely polytheistic idolaters for centuries not until when Mohammed came with his concept of monotheism. This aspect of history justifies my assertion in the previous explanation about pre-Islamic Ishmaelites who were purely idolaters.
Allah was regarded as the Chief god among the 360 gods worshipped in the Ka'aba (that black box you see them bowing to) in Mecca. Islamic theologians believe that Abraham and Ishmael built that Ka'aba. How could they have done that? Allah was an idolatrous deity worshipped by the Quraysh tribe which Mohammed belonged to. In fact, they believed Allah (the god) had three daughters: Al-at, Al- Uzza, and Al- Manat. When Mohammed came with his monotheistic concept of God, he fought idol worshippers in Mecca and Medina, he pronounced Allah as one true God and rejected the idea that he had children.
IF ALLAH IS A LOCAL DEITY OF THE ARABS, WHAT IS THE ARABIC NAME FOR THE SUPREME GOD (THE FATHER OF OUR LORD JESUS CHRIST) IN ARAB LANGUAGE?
Till date, no one has come up with an alternative name to God in Arabic language. I believe every nation on earth have a name they call God in their language, so what do Arab Christians call God??
Arab Christians call God Allah, they refer to the father of our Lord Jesus Christ as Allah. No other name for God in Arabic language except Allah.
THE CONCEPT OF A SUPREME GOD IN DIFFERENT TRIBES AND TONGUES
I pray you understand what I'm about to say because it may rattle you a bit - but if you read it with an open mind, you will grab it well. Actually, God has no name in earthly tribe/ tongue.
What every tribe and tongue of the world have is a ‘concept of Supreme being’, that is, every tribe in the world believes there is a supreme being that created the world and made all things. Even pagans believe that there is a supreme deity that created the universe so every nation decided to name that supreme being in their mother tongue. Some call Him God, 'Olorun', Chineke, Ubangiji, 'El', etc. God didn't reveal Himself to any man using these names but because man couldn't find a perfect name that suits the Supreme Being, they called Him what they believe He is. 
In fact, traditional worshippers believe that there is a Supreme being but the idols like ' Sango', 'Amadioha',etc. are his messengers which is opposed to the Christian faith. They tried to replace Christ with idols. The bible said Christ is our mediator. So in concept, every nation and world religion believe in a Supreme Being called by the names they have chosen. 
In fact, God Himself didn't reveal His specific name to Moses.  When Moses asked Him who he should tell Pharaoh sent him, God said I AM WHO I AM (Exodus 2:14). The Hebrew word 'YAHWEH' means I AM THAT I AM. It is like telling Moses that, none of your language can describe me well, just accept me as I AM.
The word ' JEHOVAH' is a Latinized pronunciation of 'YAHWEH'. Hebrews don't pronounce letter 'J'. It is best pronounced as 'yehovah' or 'Yahweh', which means I AM THAT I AM. God didn't give anyone His specific name. That's why we sing: 'You are bigger than what we call you'.

LONG BEFORE CHRISTIANITY CAME TO AFRICA
As a Yoruba man, I discovered that my people call God ‘Olorun’ (owner of heaven) even before they were evangelized by the missionaries. When Rev Thomas Birch Freeman stepped his feet in Lagos in 1842, he met a people who were totally idol worshippers but who had the belief that there is a supreme 'god' (Olorun) among other gods like Sango, Esu, Obatala, Ifa, etc. They believe he created the heavens and rules from there.
So the missionaries didn't have any other option than to accept 'Olorun' (Owner of heaven) as the Supreme God who is the Father of our Lord Jesus Christ. They also call him 'Olodumare'. The etymology of that word is 'odu'- an 'Ifa' personality that is supreme. So they called 'Olodumare'. However, 'Olorun' (Owner of heaven) was generally accepted and used in the bible. Do not forget that the name 'Olorun' did not start with Christian missionaries but idol worshippers who do not have a proper revelation of God but believe there's a supreme being called 'Olorun'.
Even the English word 'GOD' is not what God called Himself to the English people, they also have a concept of a Supreme Being. They got the name from a German word 'gott' to mean the Supreme Being that created heaven and earth but no English man has ever seen or heard 'God' declaring  His official name as 'God'. It is a 6th century invention in the Codex Argenteus. It was not so in Hebrew language.
That tells us something about God. HE IS INDESCRIBABLE WITH HUMAN LANGUAGE BUT HAS ACCEPTED THE NAMES WE GIVE HIM.
LIKE PAUL IN ATHENS
In Acts17:23, “For as I walked around and looked carefully at your objects of worship, I even found an altar with this inscription: TO AN UNKNOWN GOD. Now what you worship as something unknown I am going to proclaim to you.”
When Paul got to Athens, he saw an inscription on their altar: TO AN UNKNOWN GOD. When he saw that, he knew that they believed in a Supreme Being but they did not know him. Look at what he said: 'Now what you worship as something unknown I am going to proclaim to you'. Hallelujah.
That was what happened to all tribes of the world, prior to the coming of Christianity, every tribe had a name they were calling God which was adapted by the missionaries into our local bibles as the name for the Supreme Being- Father of our Lord Jesus Christ. Do you understand?

BACK TO THE ALLAH DEBATE
Since Arabs believed there is a Supreme God who is superior to the gods in the Ka'aba, the missionaries also were left with no other option than to call God- the Supreme Being, 'Allah' 
SAME NAME BUT DIFFERENT BELIEFS AND UNDERSTANDING
If you ask traditional worshippers if they believe in God, you can be 100% sure they'll say 'Yes'. If you ask them what they call Him, you'll be surprised that they'll call Him a name in your local bible, Chineke, Chukwu, Oluwa, Olorun, etc.
But despite the fact that the name 'Allah' sounds same in the mouth of Muslims and Christians, Arab Christians refer to the Father of our Lord Jesus Christ and they believe He's triune. Allah of Muslims is not the same with Allah of Christians. Although they are the same word but called with different understanding. Just as an Ifa priest would call 'Olorun' believing that 'Olorun' sent Ifa, Sango etc., I will call the same 'Olorun' with the understanding that He sent Christ. 
We would not be wrong to say our own 'Olorun' is not the same as theirs because His beliefs and principles differ from theirs. He revealed Himself to us through Christ. Can it be the same?
DO WE SERVE THE SAME GOD WITH MUSLIMS?
NO. Their Allah told them to kill infidels ((Surahs) 9:5; 2:191; 2:193; 3:118; 4:75,76; 5:33, 8:12; 8:65; 9:73,123; 33:60-62.) and he is not triune in nature. My own God (Allah in Arabic) is triune and loving. He sent Christ to die for my sins and He gave me the message of His love to sinners, not to kill. So is it the same 'Allah'?