All female mammals have clitori and so are capable of orgasm with sufficient stimulation and arousal/excitement. I am not sure about other sorts of animals. The clitoris is a residual embryological penis with similar nerves.
Many sorts of mammals use sex recreationally. Monkeys (and apes like us), especially. A blast of oxytocin and various endorphins is a nice thing regardless of whether it is a part of bonding or just for fun. For non-humans - and most humans now, it is not purposeful reproduction but a simple biological urge like thirst. Hence masturbation in various forms in mammals including dogs.
It's a shame that so many female humans find orgasm so elusive or difficult to reach whether in an exciting new relationship or an old one. I have spoken with many women of all ages about this. (Yes, women in their 80s like sex.) While a lucky few are able to orgasm at the touch of a penis, most not. It depends on level of attraction and arousal - the psychological component; and stimulation - the biological component. Re the latter, I sometimes recommend vibrators. Why not?
However, this is not a sex therapy post. I am skeptical about this theory: Why do women orgasm?