As a company led by two product people, we’ve definitely had a bias towards product management as a team. We’ve even hired product people to take on roles in customer facing and sales roles here! Fortunately, good product management practices can help just about any area of the business - after all, what process can’t be measured, iterated, and learned from? Personally, I split my role now, and find myself in all areas of the business, but I’ll always have a special place for product activities. I have to make sure that I don’t swoop in and overly influence the team, as frankly, they are more capable than I ever was as a product manager.
I love talking to our customers! As a matter of fact, my calendar is open for roadmapping clinics (Book Your Free Roadmap Clinic) and demos (Check out a ProdPad demo or book a deep dive with an expert) and whatever other conversations come my way.
Our team takes dozens of calls every month. Personally, I usually end up with 3-4 a week nowadays. It’s fewer than it was a few months ago as we’ve just hired our first person in the US (everyone, go meet and connect with Wes Galliher! https://www.linkedin.com/in/wesgalliher/ 
) and so he’s now doing the demos in the US timezones. Up until late last year, I was doing a lot of demos in my evening here in the UK, to keep up with demand from our US market!
I’m glad you ask about product analytics. As for tools, we’re pretty tool agnostic. We’ve used a bunch, and have a whole stack that would take some time to describe, but we started simple, with some home-brewed analytics, and have gone from there. One of the best things we did as a business was hire a dedicated Data Analyst on to our team, back when we were still about 12 or so people in the team. If you don’t understand how people are using your website or your app, you can’t serve them well, so invest in these areas as soon as you can.