Neither of these two words is a name of a god, which normally does not matter, because both the Christians and the Muslims believe that there exists only one god, so it is not necessary to have a proper name for that god.
Obviously, whenever a Christian believer or a Muslim believer speaks about the "God", they speak about the only god they believe in, so it makes no sense to ask them which god they mean.
A non-believer may speak about the God of the Muslims and about the God of the Christians, to make clear the intended meaning, and the latter phrase might be translated into Arabic with words meaning "the Allah of the Christians".
I'm a non-arabic speaking Muslim who prefers in non-liturgical context to use Khuda and have occasionally been "corrected" by well meaning folk.
Remnants of that time can still be seen in the Christian/Judaic Bible, where the plural of the cognate Hebrew word, i.e. "Elohim", can be encountered, though even there it has acquired in many places a singular meaning.
How do you distinguish between "having a different God" and "having different beliefs about the same God"?
But if we both claim to know a guy called John Brown, but I say he's kind, generous, hard working but a little dim and you say John Brown is hedonistic, self-entered and brilliant, are we talking about the same person?
Normally if our descriptions are similar enough that we can chalk it up to perspective, we might think it is. But if our descriptions of the person become strongly contradictory, we must conclude that we are talking about a different person.
Now if we both claim that there is only one John Brown in existence, then we might be willing to overlook slightly more of the inconsistencies, because the alternative is that one (or both) of us is misguided/wrong/a liar.
I don't think this example really is catching the core of discussion about Allah (The God). Because we both know a president who bombed two cities and by nuclear bombs. Some people will call him a war hero because he ended WWII, Some people will say he is a war criminal. You will always find something like this when you are talking about a human. Maybe it is also not just different opinions, but different access to information can give us different view on the same person. But we agree that we are talking about the same person, I just find this person to be different from what you think he/she is.
Things are totally different from religious point of view. The discussion between Christians and Muslims is mainly about the nature of The God, not that there is "The God". This also might extend to Judaism, but I don't know much about it.
So like you say, if the overlap between your A,B,C,D and my A,B,C,D is sufficiently small we should probably acknowledge that it's not the same entity. See also: Ship of Theseus.
This is the equivalent of translating "Praise Jesus" to "I love god and hate abortions"
Also, Allah to Arabic native speakers can be used to mean "The Lord" in general. Levantine Christians and even some older Mizrahis will use Allah in daily parlance for "God" or "Lord". For example, in colloquial Hebrew, Yalla/Ya Allah is a common expression similar to "Oh God" or "Jesus" or "Come On" when annoyed.
> "Alhamdulillah" [...] is used by Muslims nowadays as an alternative to Allah u Akbar, Takbir, or Narai-e-Takbir because those 3 terms have been co-opted by Jihadists.
That's just flat-out false. Allahuakbar is a phrase of critical religious significance and tied to one of the pillars of Islam (Salat). Alhamduillah has always been a common phrase. Both are used daily by Muslims globally, even those who are not religious. They are far too deeply rooted in the creed and traditions of Islam to be able to be co-opted.
It's like asking whether someone means cake or gateau. Islam is an Abrahamic religion which means they worship the same deity as Jews and Christians.
Here is a page from the Arabic website for the Holy See (Vatican). Just go to this page and search for الله :
https://www.vatican.va/content/francesco/ar/encyclicals/docu...
To take an English example, imagine a Christian saying "God is great", with an option to translate it as either: "God (generic) is great"; or "The Christian God is great".
Sure, he was probably thinking of the Christian God when he spoke, but making that explicit in the translation still changes the meaning.