I heard the phrase from a person who is not a Christian, speaking about Christmas. I wondered what he could mean other than some version of the Christian holiday, though he would not likely credit that. He would be more likely to say that it was all of the holidays of that time. Yet Hannukah is one of those classic Jewish holidays of "the bastards tried to destroy us but we're still here." A fine sentiment, and bringing in the custom of generosity to children is nice. But it is about tribal solidarity. The winter solstice does not commemorate any tradition of good will, forbearance, or generosity. I had an office-mate for fifteen months who was a Wiccan (and turned out to be a distant cousin), a lovely warm person, who would likely say in a correcting tone to all those Christians, that Wiccans teach those things all year, not just in December. Yet that is evasive of the point. It is also not true, but I suppose it theoretically could be true, so I let that part pass. You might stretch a point that the Ujamaa value of Kwanzaa, cooperative economics, has something about community-building in it, but it's not the same thing as a more universal brotherhood and generous spirit.
So the spirit of the season derives entirely from the Christian holiday, even if those values are present in other beliefs. However, this gets complicated, as that kindliness, generosity, forbearance aspect is a mostly western presentation of Christmas, and not central to the specifically religious observance. Again, I am not saying that Eastern Christians don't have those virtues, it's just that they are not tied so closely to Christmas. So Spirit of the Season is more a culturally religious or even secular value derived from the Christian holiday.
If the idea is rather a muddle in that gentleman's head I suppose I can't fault that, as the idea is rather a muddle. Just not the muddle our culture would currently credit. A different muddle.