Let's break this question down a bit.
"Disease" is a broad term to refer to any condition that impairs the normal function of the body. It can be broken down into different groups - acquired, congenital (born with it), genetic, latrogenic (side effect of treatment), idiopathic, mental, infectious, non-communicable, lifestyle (just to name a few).
So depending on what type of disease you're looking at, it can affect different types and species of organisms. If you really want to understand disease that can be transmitted between humans and animals, look into zoonosis. Even if the same pathogen is infected in different species, they may cause symptoms or be asymptomatic in others. For example, the SARS virus can be happily carried by bats without causing any disease in them due to their much higher body temperature. However, when it infects humans it causes a huge immune response (cytokine storm) which ends up killing the host. If you want to understand non-transmittable diseases between human and animals but want to understand the differences in disease presentation, I think diseases such as cancer, mental health or auto-immune disorders will be interesting to look at.
I haven't seen it mentioned yet but there are also prions (misfolded proteins) and mental health.
Also "attack" isn't the best use of word in this context when you're talking about something so broad. Yes, you can use it to describe infectious disease or auto-immune disorders but if its something like depression the cause may be a chemical imbalance, so it is not really an "attack".
I'm not all too familiar with disease that occur just in animals as I'm not a vet student (yet). I studied a lot on infectious disease and zoonosis, with a human-health perspective.