You did your son a disservice. Christmas wasn't derived from pagan holiday or celebrations.

https://www.catholic.com/magazine/online-edition/refuting-the-pagan-roots-of-christmas-claim

There are some good discussions on Quora about this with both Islam and Judaism. The short answer is, yes, both have overlap with pagan pantheistic holidays and rituals.

I do know that Taoism and Budism had one upmanship fights adjusting their religion to maximize followers for a number of years.

It's a bit over simplistic to say that Christmas is a repurposed pagan festival, but given the age of your child over simplistic is fair enough.

As a Muslim, we believe similar of what you’ve spoken, Jesus wasn’t born in December. He was born in spring. We don’t celebrate his birthday. Although we do love him as he’s a prophet. We also believe in the day of resurrection and that he will return.

It sounds like your information comes from the section of Reddit where resentful atheists figure out ways to diminish whichever religion they’re targeting at the time.

1Judaism is a closed religion, meaning you are either born Jewish or not. You cannot just practice it. Jews do not try to convert people. It's confusing because religion is a Christian concept, but Judaism is actually an ethnic religion. Think of ancient Greeks, they had their own beliefs, calendar, rituals, etc. Judaism is the same. We have our own calendar which revolves around the moon (much like China). We also follow the land or agrarian rituals, so many of our holidays revolve land.

2. It was the first monotheistic religion, which means at the time, instead of having 2 or 3 G-ds we had 1. Our main texts are the Torah (old Testament) and the Talmud (laws and rules). Essentially, you know the 10 commandments, Jews have over 400 commandments we have to follow. Most of our holidays are about celebrating times we were nearly all killed, or when our temple was destroyed. We are very family based and Most families get together every Friday night for Shabbas dinner.