Nope, doesn’t offend me, but don’t be surprised if I occasionally answer, “Thanks, and a Happy Solstice to you!”

. I don’t find “Happy Holidays” or “Season’s Greetings” offensive either and have never had a preference for one over the others, nor did I when I was still trying to make myself believe I could be Christian. There are much bigger things to worry about than that someone is wishing you happiness.
I also don’t ask folks to see their citizenship cards before saying “Happy 4th of July”.

Many Americans celebrate a specifically American secular holiday of Christmas in addition to or instead of celebrating it as a religious holiday, so I don’t see the big deal. We celebrate the American secular version of Christmas in addition to our religious holiday of Winter Solstice, for instance. We have a religious observance and exchange gifts on Winter Solistice (Dec. 21/22), then we have a Santa stocking on Christmas morning and visit relatives who celebrate Christmas as a religious holiday. If we had relatives of other religions, we would likely visit them on their holidays as well, if we were invited.
If I know someone well enough to know their religion, I will try to offer appropriate greetings for their holidays (and try to be sure they actually
are on those holidays, rather than assuming that all other holidays fall on Christmas).
Freedom of religion does not mean that you will never have to acknowledge that another religion exists.
What I do find ridiculous is the bit about “ho, ho, ho” in Australia
news.yahoo.com/s/afp/20071115/wl_asia_afp/lifestyleaustraliachristmasoffbeat
and “family trees” at an American hardware chain
snopes.com/politics/christmas/familytree.asp.
Neither is quite as annoying, however, as the perennial lack of understanding shown by saying that Xmas is trying to “take Christ out of Christmas”. That is simply lack of understanding the history of one’s own religion.