I think some sort of animal safety/awareness/responsibility should be covered in various topics through primary and secondary.
In primary, simple stuff like what a growl means (little children do not know this) what a bared teeth snarly face means (again little children often think this is a happy smile).
Moving on to things like not approaching animals you do not know that do not have an adult with them, whether we believe the person with the dog/cat/rabbit/horse when they say the animal is safe because some folk do NOT know...
In secondary, incorporate it into other subjects - for example PSHE - History, Economics - you can cover irresponsible breeding, the dangerous dogs act, why people buy dogs to look 'hard' (often young kids in their first bedsits will buy a 'staffy type' puppy as its cheap, easy to feed, makes them feel more secure), what the dangers are.
Dogs are a huge part of our society, this country trades to a great extent on its 'animal loving nation' and yet our education covers next to NOTHING.
I don't see anyone saying that children are to blame when dogs bite or kill, even if they are actively abusing an animal.. an ADULT is responsible for the animal AND an adult is responsible for the child!
I do think if education starts with children, from a young age, we stand a better chance of avoiding young people buying dogs to look intimidating or to use as weapons, avoiding 'just one litter' back yard breeders, avoiding the use of aversives as training methods which often results in a dangerous and unpredictable animal.
I doubt this will ever happen, it needs a model of education that works practical stuff into the curriculum and we don't currently have that!