Jesus was not a Christian and Buddha was not a Buddhist
There is a saying among non-religious spiritual folks that "Jesus was not a Christian and Buddha was not a Buddhist" which I agree with. It basically means that Jesus and Buddha both taught similar simple spiritual truths about loving others, keeping moral values/integrity, cultivating a spiritual life, transcending the material world and the ego from within, losing one's sense of self/ego to merge with God and others, and shifting awareness from egocentricity to cosmocentricity (in other words, shifting from focus on oneself and one's individual needs to those of the collective interconnected universe, aka "God"). However, their FOLLOWERS are the ones who changed their leader's original teachings, gradually developing them into the religions we have today.
Thus, theoretically, if Jesus and Buddha were around today (physically on Earth), they would probably not agree with the modern versions of the religions attributed to them. For example, Buddha originally taught that we should not make idols or worship them, yet many of his followers today, even in the different Buddhist sects, do make idols of Buddha and worship or pray to them (even though they try to deny it by trying to claim that they are just "focusing" on him, yeah right). And Jesus as depicted in the earliest writings of the gospels preached simple messages of love as the way to God and Heaven, rather than the atonement teachings that "You have to believe in me to be saved. You have to repent for your sins and believe that I died on the cross for you to be saved and get into Heaven." etc. which his followers set up later (e.g. Gospel of John and Paul's Epistles) that have become canon today.
But suffice to say, neither Jesus nor Buddha wrote anything down themselves, so all we have is second and third hand testimony of what they taught, mainly from anonymous sources, even if some of it is consistent and agreed upon. I don't have to tell you though, that humans are fallible, and that the authors of the Christian and Buddhist scriptures are privy to the same imperfections and mistakes that you and I are.
These are examples of how our religions evolve over time, molded by humans for their own agendas and colored by their perceptions, even though they may begin from spiritual teachers with good intentions.