Do Christians really believe that Muslims worship a Moon god ?

They worship a false idol.  They do not worship the God of Abraham, since Jesus is God and Muslims deny that Jesus is God, so it cannot be said that Muslims worship the God of Abraham.

Christians may or may not claim that Muslims worship a moon god.  I don't focus on that.  But as I wrote, Islam does worship a false idol.

I don't believe they do.  I do, however, note that there was a pre-Islamic Moon goddess worshipped particularly in Mecca called Al-Lat, that Ramadan and the start of the day is marked by observation of the crescent in the sky and that it's used as a symbol of Islam.  It's simply that the origin of the faith, like Christianity, is not hermetically sealed off from other spiritual traditions and that's not a problem.

Because, despite Muslim claims, Allah is nothing like the one, true God that Christians worship.

Read this article

Is Allah the God of the Bible, or is Allah the moon god of ancient Arabia?

https://www.billionbibles.org/sharia/allah-moon-go...

Christians believe in the God "I AM"

Muslims believe in the God "Allah"

"I AM" and "Allah" SAME GOD.

Jews, Christians, and Muslims all worship the God of Abraham. The statements which say it is a different God come from Fundamentalists who are trying to prove they are the only ones with truth and everyone else is some sort of heathen or heretic. 

These are the same people (pastors mostly) who I used to hear as a child say that the X in Christmas (Xmas) was a plot by Jewish merchants to take Christ out of Christmas.  

All of religion is completely absurd 

A couple of Christian pastors made that story up, and ignorant Christians everywhere spread it far and wide.  They love to put others down.

Muslims 

god does not claim Jesus as his son. So clearly their god isn’t the God of the Bible.