Short Answer
The answer to this is that theologically Chalcedonian (Trinitarian) creed believes in Jesus's divinity hence practically they don’t worship the same God as Muslims do. The Quran explicitly calls the Christians disbelievers on account of taking Jesus as God.
Surely, disbelievers are those who said: "Allah is the third of the three (in a Trinity)." But there is no Ilah (god) (none who has the right to be worshipped) but One Ilah (God -Allah). And if they cease not from what they say, verily, a painful torment will befall on the disbelievers among them.
Al-Maeda, Chapter #5, Verse #73
But there are a few minority Christian sects like Universal Church, Mormonism etc which don’t believe in the divinity of Jesus and hence worship Allah who revealed the Injeel (Bible). Jews worship the lord of Bani Israel which in none but Allah hence they worship the same God as Muslims. Moreover Jews themselves consider Christians to be idolaters (Avodah Zarah) while they consider Muslims to be monotheists.
Long Answer:
First we need to understand what is God before judging whether we worship the same God or not, Allah clearly defines God in Surah Ikhlas:
Say: He is Allah, the One and Only; [1] Allah, the Eternal, Absolute; [2] He begetteth not, nor is He begotten; [3] And there is none comparable unto Him.
Hence any religion which violates the above definition of God does not worship Allah but their own model of Allah. Only Jewish religion is compatible with the above definition of God.
Theologically Chalcedonian (Trinitarian) creed believes in Jesus's divinity hence practically they don’t worship the same God as defined in Surah Ikhlas. The Quran explicitly calls the Christians disbelievers on account of taking Jesus as God.
Surely, disbelievers are those who said: "Allah is the third of the three (in a Trinity)." But there is no Ilah (god) (none who has the right to be worshipped) but One Ilah (God -Allah). And if they cease not from what they say, verily, a painful torment will befall on the disbelievers among them.
Al-Maeda, Chapter #5, Verse #73
But there are a few minority Christian sects like Universal Church, Mormonism etc which don’t believe in the divinity of Jesus and hence worship Allah who revealed the Injeel. Jews worship the lord of Bani Israel which in none but Allah hence they worship the same God as Muslims. Moreover Jews themselves consider Christians to be idolaters (Avodah Zarah) while they consider Muslims to be monotheists (Rambam in Sefer Qedusha, Hilkhot Maakhalot Asurot 11:7 ) having share in the hereafter.
Many people would cite the following verse to claim that Muslims and Christians worship the same God, When Allah says the following:
Say, "O People of the Scripture, come to a word that is equitable between us and you - that we will not worship except Allah and not associate anything with Him and not take one another as lords instead of Allah." (Surah Al-e-Imran, verse 64)
The truth is that it just calls the Ahle Kitab (People of the Books, Bible, Torah) to the absolute truth which is in their scriptures without regard to their actual beliefs, Moreover Allah just calls the Ahle Kitab to worship one God alone which they have ceased doing. Although Jews worship the same God theologically, the Ahadith (Saying of Prophet Mohammed – May peace be upon him) say that they have taken their Rabbis as God by blindly following them. The prophet (pbuh) has equated their blind following of Rabbis with polytheism:
Almighty Allah Says:
“They (Jews and Christians) took their Rabbis and their monks to be their lords besides Allah (by obeying them in things wish they made lawful or unlawful according to their own desires without being ordered by Allah), and (they also took as their lords) Messiah, son of Maryam (Mary), while they (Jews and Christians) were commanded (in the Torah and the Gospel) to worship none but one Ilah (God i.e. Allah), La Ilaha Illa Huwa (None has the right to be worshiped but He). Praise and glory is to Him (for above is He) from having the partners they associate (with Him).”
(V. 9:31)
Once, while Allah’s Messenger (P.B.U.H) was reciting the above verse, ‘ Adi bin Hatim said, “O Allah’s Prophet ! They do not worship them (Rabbis and Monks).’ Allah’s Messenger said, “They certainly do. They (i.e. Rabbis and Monks) made legal things illegal, and illegal things legal and they (i.e. Jews and Christians) followed them; and by doing so they really worshiped them.”
(Narrated by Ahmed, At-tirmidhi, and Ibn Jarir). (Tafsai At-Tabari, Vol. 10, Page No. 114)
