OP - cats can be taken out for walks too. I have a harness for my cat and used to take him on walks around my building. It's easier to start the cat young but you can also slowly train older cats to get used to a harness and then a harness+leash, and then walking around your home, and finally actually taking longer and longer walks outside the door. It's not quite like walking a dog since cats tend to go very slowly and meander all over the place but people will stop and talk to you for sure.
You're also completely right that practicing socializing makes it more comfortable, like any other habit.
I moved to a new place a couple years ago where I didn't know anyone and wanted to figure out how to make new friends. I usually test as an introvert by MBTI and hardly thought of myself as a mover-shaker. But I'm also a scientist so I tried an experiment where I brainstormed the actions of a social organizer and executed them. I met people through meetup/OKC/work/hobbies (one of my friends used craigslist to great success) and got the contact info of people I liked talking to and then invited them to hang out individually or in small groups for low pressure get togethers (like coffee/tea, lunches, happy hour, potlucks, cooking meals, biking around town, walks, hikes). In the beginning, this felt very awkward but I just acknowledged the feeling and kept practicing the actions, while noting what seemed to work and what didn't.
I was surprised at how many people love being invited out and said yes. From what I've observed, most people (esp. singles) want to do social things but are waiting for someone else to make the first move or to organize. For me personally, it takes time and many one-on-one or small group interactions to make the kind of close friends you described (the ones you can call up to talk or to hang out regularly). These repeated hangouts may be uncomfortable at first but it's investing - both for increasing your socializing comfort level and for creating friends. Over the two years, I built a couple of close friendships and found several close groups that I have fun hanging out with.
Best of luck to you, keep practicing!