I feel happier around animals. I have always been interested in and felt connected with animals and nature, I think. When we were living in our old house which had small forest next to it, I used to take snails and tortoises from the roads so they won't get killed by the cars. I remember, when I was feeling anxious, I was just going outside and try to find them and watch them without hurting.
I've been interested in cats for 2-3 years (the times I have). They were mostly afraid of humans in the places I've lived so I've always found them as distant, cute, fragile animals. It was a big chance to pet one so I have more sympathy for them than dogs (dogs seems stronger). At our old house, there was a little dog that always came to me and didn't leave me when I wanted him to when I was very young so I'm still a bit scared of them but I started to be more comfortable around them recently. I like birds, too.
Yes, I find those animals to be helpful when I feel anxious and I think I'm a bit more relaxed during some social situations for two years with our pet but I can't say for the more anxiety giving ones. I also think it depends on people, like how much they have connection for animals and nature in general.