A perennial question that pop up almost every year during Christmas, however for me, the question never popped into my mind. For me it is a given that there is no problems with Buddhists celebrating Christmas, going to mass or even singing along during services. I believe that this only becomes a question when people see Buddhism as a religion rather than a system of teachings.
I remember quite a long time ago, I was invited to a Christmas mass with another Buddhist friend, I was singing enthusiastically all the gospel songs and Christmas carols, but I noticed that the friend was hesitant to sing or perhaps he don’t really know the tunes well. Either way, for me even if you asks me to visit a mosque or Hindu temple and pray together with the believers, I will do it as well as any of them can and as enthusiastically also. I don’t believe that doing that makes me any “less Buddhist” or makes me believe in anything different.
More fundamentally, we can see Buddhism as a set of teachings on how to live effectively with ourselves and others. The Buddha never claimed to be saviour, neither did he claimed to have invented anything new nor got any message from anyone. Instead, he spoke of how these laws and teachings already exists in nature and he only discovered them through wisdom, insight and meditative practises. Anyone who had gone through a similar training would be able to develop the same insights he has on the phenomenon of nature and the universe.
As such, the Buddha is seen as a great teacher and not some deity or godly figure. The only reason why people start to see him as more godly and worship him is due to the distance away from him and a lack of insight. Just imagine meeting say a forest dwelling native who never seen technology and tell the person stories of a man who can make metal birds fly, the native will think this engineer must have godly powers and may even start worshipping an image of him! The same is probably true for those who cannot see the true nature of worldly phenomenon and sees all the achievements of the Buddha as godly when he himself could probably realise the same truth when putting in the right effort.
Secondly, Buddha did not set out to create another religion, this was invented by people over time. It is just a set of teachings about the nature of the world, just like you don’t call super string theories a religion, either is the set of teachings about the nature of suchness. And also because its about natural law, one don’t have to be a “Buddhist” to discover this laws of nature. Of course, this doesn’t make every other religion a Buddhist teaching nor necessarily compatible with some of its teachings neither; there any ground rules to reject teachings from other religions (and when bitching about them lolz. Sorry this my own opinion, not in the teachings!)
Lastly, he did not seek to convert anyone to this new system of thinking and practices. In some of the Buddhist texts, we can read about how the Buddha encourages fellow Brahmins and their followers to practise even better within their own system of teachings. Of course, the problem here is, for a lot of them, after hearing the arguments put forth by the Buddha, many of them decided to following him instead! That said, Buddhism is never a teaching of mutually exclusive practise, it encourages fellow Buddhists to show respect to great religious teachers of love and peace, at the same time to share with others how to better practise love, kindness, joy, generosity and equanimity within their own system of teachings whenever possible.
So for Buddhists, whichever religious festival one are attending, one should not fear losing one’s own self-identity nor believe. If you need to sing hymns, say any prayers or perform any rituals (which does not involve harming or killing, of course), just do it and be present at the moment. Instead of being a lesser Buddhist, this is exactly what a Buddhist should do.