This depends on the company's norms and how different teams have been set up. I've seen user research lead this entirely, and I've also seen a dedicated "Voice of the Customer" program manager who wrangled all internal listening channels.
One area where I've seen PMM historically drive a ton of value is with market and competitive insights. Especially at Gong, given our product, Product Managers have access to customer insights but oftentimes don't have visibility into opportunities outside of the customer base.
Focusing specifically on customer feedback (or user feedback), PMs generally own collecting and synthesizing feedback during the design partner and beta periods. After the release - it depends. The Product team should own end user feedback post-launch on a systematic level; where PMM can add value is staying close to the field and new segments you're prioritizing on the GTM side.
For example, if the GTM team is going after a new industry, PMM should be listening to product feedback from these new customers and potential customers separately and differently from how we'd treat feedback from core end users. What about new personas or teams? For these high priority segments, the PMM org should be packaging up their feedback, market requirements, and market size to inform product roadmapping.
Yes depending on the type of feedback sought we have different owners, but we are trynig to move to a place where the Customer PMM owns tracking of outreach (we don't want to ping the same account 7 times across different deparments).
In-product NPS is owned by customer success, but interviews off the back of low or high scores are often handled jointly by CS and PMM.
Beta feedback is owned by the product team, but PMM might often tag along for interviews so we can see from a customer's perspective how they would describe the value they receive.
Product UX feedback is owned by design (sits in product). Website feedback is owned by Acquisition Marketing.
Testimonials and case studies are owned by Customer PMM
Value-fit and pricing interviews are owned by PMM