One of the criteria that historians use to determine the authentic sayings of Jesus from the Gospels is something called double dissimilarity. Double dissimilarity basically means they look for sayings in the Gospels that don’t look too Jewish or too Christian. If it is dissimilar to both Jewish and Christian teachings, then it probably goes back to the historical Jesus.
For example, take the title of the Son of Man that Jesus uses in the Gospels. The idea of the Son of Man was not a major image in Judaism and it did not become popular within Christianity. Therefore, the self-description of the Son of Man probably goes back to the historical Jesus.
But there are two problems with this.
The first is that Jesus was Jewish. He was raised in a Jewish context, surrounded himself with Jewish disciples and preached to Jewish crowds, often quoting the Jewish scriptures. We would expect that authentic sayings of Jesus would have something in common with Judaism.
The second is that Jesus was the founder of Christianity. The church emerged out of the group of disciples who directly followed Jesus. The sayings of Jesus were the foundation of the Christian faith. We would expect that authentic sayings of Jesus would be picked up by the early Christians.
Is this some sort apologetic defence of Christianity? It could be used in that way, but my observations come from my historical interests.
Let’s look at this in a similar but different context.
The person we know of as the Buddha was originally known as Siddhārtha Gautama. Gautama was raised in a Hindu setting and eventually became the founder of Buddhism. We would expect within the recorded teachings of the Buddha, sayings that overlap with both Hindu and Buddhist teachings. This is exactly what we see.
In the same way, a Jewish Jesus who was the founder of Christianity should have said things that sounded both Jewish and Christian.
I personally do not find the criteria of double dissimilarity to be particularly helpful in studying the historical Jesus.