Except that the majority of Christian sects including the Catholics and Methodists/Presbyterians/lutherans/etc say the Old testament is superseded by the New, and is not needed to be followed literally as the Jews do.
And the new is far more turn the other cheek than it is burn the fornicators.
Meanwhile the Koran is ALL to be followed by a good muslim, including the bits about the only non-muslim being a slave to a muslim or dead.
The closest thing to a New Testament for Muslims is the Bahai religion, which grew out of a splinter from the main Islamic religion by a muslim prophet who said the Koran was obsolete and a new way was needed, much the way Islam grew out of a Christian prophet who thought Christianity needed a change.
Needless to say, strict Islamics in the Wahabi mould like killing Bahai even better than they like killing Christians.
The first Moslems were mainly Monophysite Christians in the near east and north africa, the dominant christian dogma of the timewas Dyophysite. The view that Christ was merely a prophet tied in nicely with their view of Christ having a single nature, in contrast to the dominant view of Christ having both a divine nature and a human nature in one soul.
Given the theological debate of the time involved stoning those of opposing views, or bashing their heads in with staves in street fights, it must have been rather a relief to follow a schismatic preacher into the wilderness. Followed by several generations of converting the tribes and their descendants coming back leading armies which got as far as most of spain and deep into the transylvanian mountains.