You're attacking a geographical interpretation of 'western' which I don't think is the usual meaning.
Rome and its successor states colonised western Europe over the course of ~2000 years. Hence the culture unique to the 'West' is in part that which was unique to the Romans. 'Western' doesn't so much describe culture which was indigenous to western European countries (much of which is lost) nor is it a catch all term for cultures which were influenced by the Romans to a lesser degree. It describes the cultures most heavily influenced by the colonisation of the Western Roman Empire and its successors over the longest period of time.
If Western Europe had been colonised by the Islamic Empire (in the way that the Eastern Roman empire was) then 'Western' would indeed include Islamic culture.
Britons and Danes were not Roman, their culture in antiquity was very different, yet they were gradually influence/coerced into adopting many of its cultural practices. Modern Britain and Denmark are therefore 'Western', even though e.g Persia had far more in common with the Romans/Greeks at the time. The culture of Turkish people is heavily influenced by ancient Greek and Roman colonisation, and hence they are 'more western' than other areas in the near east. Yet later conquests by Arabs and Mongols bought a much more diverse set of religious and cultural practices. Similarly, regions like Greece itself and Southern Spain have been colonised by eastern empires, and are somewhat 'less western', though they were later reintegrated into the 'West'.
Rome and its successor states colonised western Europe over the course of ~2000 years. Hence the culture unique to the 'West' is in part that which was unique to the Romans. 'Western' doesn't so much describe culture which was indigenous to western European countries (much of which is lost) nor is it a catch all term for cultures which were influenced by the Romans to a lesser degree. It describes the cultures most heavily influenced by the colonisation of the Western Roman Empire and its successors over the longest period of time.
If Western Europe had been colonised by the Islamic Empire (in the way that the Eastern Roman empire was) then 'Western' would indeed include Islamic culture.
Britons and Danes were not Roman, their culture in antiquity was very different, yet they were gradually influence/coerced into adopting many of its cultural practices. Modern Britain and Denmark are therefore 'Western', even though e.g Persia had far more in common with the Romans/Greeks at the time. The culture of Turkish people is heavily influenced by ancient Greek and Roman colonisation, and hence they are 'more western' than other areas in the near east. Yet later conquests by Arabs and Mongols bought a much more diverse set of religious and cultural practices. Similarly, regions like Greece itself and Southern Spain have been colonised by eastern empires, and are somewhat 'less western', though they were later reintegrated into the 'West'.