Nothing I've seen supports that. But anytime that you make your cells divide or damage your cells your cancer risk is increased a bit. Being cut for example. Anal penetration can be taxing on the body. Sometimes it takes me a few days up to a week to feel normal again after anal stretching (a big dog would require this). That is a stress on the body that probably does increase cancer risk by a small amount.
What's relevant is things that increase cancer risk to a high degree. Animal sex is not proven to be one of them.
Cancer, I say is less likely. Diseases, I can say unless you were having sex with a complete random stray dog or some stray in a third world country. But its still a low possibility.
Those articles and videos are just to help people that have no clue on zoos act like they know every bit of the dangers. When no actual research was done. Anyone can make up research articles.