[The first Pillar of Islam] is the shahada, or the profession of the faith: there is no god but god [sic] and Muhammad is God's prophet, which is sometimes translated as "There is no god but Allah and Muhammad is Allah's prophet," which tries to make Muslims sound other and ignores the fact that the Arabic word for God--whether you are Christian or Jewish or Muslim--is Allah.
Is Green right about this? My understanding is that, from a purely linguistic viewpoint, it is true that the Arabic word for "God" is Allah. However, I would point out that while Arabic-speaking Christians, Jews, and Muslims may all use the same word to mean "God," it doesn't necessarily follow that their conceptions of God are the same. In fact, while Christians, Jews, and Muslims do share some common beliefs about God--for example, that that there is one God, that He is the Creator of all, that He is all-powerful and all-knowing, etc.--their beliefs about the nature of God exhibit some major differences. For example, Muslims hold to a unitarian view of God--that He is one in essence and in person--while Christians hold to a Trinitarian view--that God is one in essence but three in person. Indeed, for Muslims, perhaps the most serious sin is shirk--assigning "partners" to God, while for Christians denying the divinity of Jesus is a rejection of a fundamental truth. Moreover, for Muslims, whatever God decrees is good by virtue of having been pronounced good by Him, but for Christians whatever God decrees is good because goodness is an essential part of God's nature.
In short, if Green is trying to argue that Christians and Muslims and Jews essentially worship the same God, that is--at least with regard to Christians and Muslims--manifestly false--even if this has become a commonly-held belief among many people, especially those who consider themselves intellectually sophisticated. Mr. Green clearly believes that many non-Muslims have misconceptions about Islam, and seems to see it as his duty to clear up those misconceptions. However, he does no one any good by implying something that is not true.
Furthermore, in at least one country, Allah is officially viewed as an exclusively Muslim word. In 2014, the highest court in Malaysia, a majority Muslim country, ruled that Christians and other non-Muslims may not use Allah to refer to God. (Incidentally, Malaysians speak a form of Malay, not Arabic, as their native language, so I assume Allah is a loan word from Arabic).
I might also note that in his video Green tells his viewers that "the angel Gabriel appeared to Muhammad"--as if this were a fact---rather than saying that Muslims believe that the angel Gabriel appeared to Muhammad. Although I understand Green identifies himself as an Episcopalian, I rather doubt he would present Christian beliefs in his videos as if they were indisputable facts. This is rather reflective of a tendency I've noticed about some non-Muslim writing about Islam--to show that they are sympathetic to Muslims, they treat Muslim beliefs with much more respect than they would Christian beliefs. This hardly seems even-handed.
Image of the Quran--the Islamic scriptures--from Wikimedia Commons