God tells us in the Quran that he has created us to worship him.
See Quran 51:56
((I have not created the jinn and mankind except to worship me.))
And the Prophet sallAllahu alaihi wa sallam said: 'Do you know what the right of God is upon his servants (mankind)?'
'The right of God upon his servants is that they worship him and they don't worship anything else.'
This is the core of Islam and it is the religion of all of the Prophets and messengers.
So a person is not truly morally upright until they fulfill God's right upon them.
To disbelieve in God and to worship other things and violate your created purpose is the greatest moral wrong.
In addition, God says in the Quran that he has given us hearts that enable to live our lives properly. Having natural morals is a blessing from God which we should be thankful for by worshiping him.
See Quran 67:23
((He is the one who has created for you all hearing, sight, and
hearts; only few of you are thankful.))
In addition, the Prophet sallAllah alaihi wa sallam said: 'I have only been sent to complete noble manners.'
He didn't say nobody had manners before he came.
But anyone who looks into the Quran and Sunnah will see that it is of a much higher moral caliber than what normal people adhere to.
In addition, a Prophet was sent to every nation of people telling them to worship God and do good actions and be morally upright.
See Quran 16:36
((We have sent to every nation a messenger telling them to worship God
and to avoid the worship of all else.))
So take the UK for example where I am from. People for a long time read the Bible and took their moral fiber from it and brought their children up with those moral guidelines. So those atheists who think that they have their morals from no religion are in part very wrong because they were raised by generations of people who were conditioned by what they read in the Bible.
To put in simply, God's messengers left behind a moral legacy that was adhered to by many even if they didn't believe.
What you are beginning to see now is a breakdown of morals in society because people don't have a set of rules to say how you can and can't act.
Also, the one who truly fears God's punishment and hopes for his reward will always be more morally upright than someone who doesn't. As long as the God fearing person is acting according to revelation (as opposed to unjust man-made 'religious' rules).