I find this idea that 'morality is subjective' very interesting. I think this is very characteristic of modernism and post-modernism. The reason I say that is that, in traditional societies, Western society included, morality was anything but subjective. It was given by 'God's law'. Everyone agreed on it (or thought they did, although if you analyse from the contemporary viewpoint there was actually a lot of conflict between various interpretations. But I will leave that aside for the purposes of this discussion.)
So as part of the rise of modern philosophy (starting with Hume, Locke, Descartes, and so on) the idea of 'the source of morality' has undergone a complete transformation. In a notionally religious society, such as European society was prior to the Enlightenment, morality was understood in terms of public statutes which had developed in accordance with 'God's law'. (Many of the vestiges of this understanding remain in common law and civil society today, although if secular activism has its way, its days are surely numbered.)
But with the separation of Church and State, the translation of the Bible into the vernacular language, and the rise of Protestantism, the basis of morality shifted from insitutional authority to individual conscience. Accordingly, religious consciousness - and notions of moral truths - have become a private matter, almost, one might say, a matter of opinion
. Hence the idea that 'morality is subjective'.
Now there is a conflict here. A very early, and important, distinction in Western philosophy was Plato's distinction between 'mere opinion' and 'true knowledge'
. The secular outlook has implicitly rejected, or at least overlooked, this distinction by declaring that issues of moral truth are subjective. As secular society has, on the whole, rejected religious, spiritual or philosophical notions of morality in favour of science, then 'truth' also is now practically limited to statements which can be verified 'objectively', or within specialised languages such as mathematics or various realms of expert discourse. So the notion of there being a 'higher truth' - 'truth with a capital T' - or some kind of moral truth that has been either 'revealed' or can be discovered through 'philosophical enquiry' is generally, although often quite unconsciously, rejected.
The upshot is that we moderns, generally without even being aware of it, have actually rejected
the traditional philosophical enterprise of 'arriving at wisdom about the true nature of things by rigourous and disciplined self-enquiry'. This has been displaced
into so-called 'natural philosophy' which to all intents and purposes is in the domain of scientific specialists, the boffins at CERN and NASA. Implicitly, we assume that the world of the senses is reality, and our sole means of learning about it, science. Which, in my view, actually leaves a gap, a vacuum
, in the centre of our being. And a great deal of contemporary 'culture' is aimed at filling it, but as we are not really aware of what is is, or how it got there, we are never satisfied.