I have a good degree of empathy for both humans and animals. It is deep and sudden.
Though in the past I've found it difficult to empathize in some social circumstances expecting empathy. For example, when someone close to me was getting married, I felt more of a sense of grief than happiness for them. I felt contrived trying to pretend to be happy for them because at that time I couldn't imagine marriage being a good thing. This may have also been related to my not being able to observe the relationship, or analyze it, as I didn't know the groom-to-be.
In retrospect I look back and wonder what I was thinking. In this way, I'd say that my younger self had not really established a socially acceptable form of empathy yet. People were kind of annoying to me then as I found them intrusive and judgemental. Now, in many cases, I value them for that same annoyance, in measured amounts. My sense of empathy and tolerance has broadened over time.
I try to remember that most pets would eat me if they didn't have any other food, and that there is a food chain. In addition to empathy, nature deserves respect. Also, in this way, vegetarian pets are probably better. If an animal is suffering, I cannot stand to watch and would rather put it out of its misery.
I instinctively protect both people and animals. At times I have reactively shielded animals from being harmed, though I've also hunted them to eat.