For me, it depends. I used to store all of my feedback so I could look for patterns, which helped me avoid catering to a squeaky wheel. However, doing so means that the feedback can sit somewhere, untriaged. I think not using feedback in the near future is worse than risking listening to one point that not everyone shares.
So now I tried to pull in some cross-discipline team members as soon as I get some feedback so we can quickly triage it: do something about it now or set it aside (and, realistically speaking, never look at it again).
If I can't or don't want to do that (e.g., it's not enough feedback to warrant bringing people together, or if I suspect the feedback is a one-off issue, or if I don't fully understand it and want to dig into that area deeper later), then I store it somewhere.
I used Excel for a while, but I didn't like that it was tough to share/collaborate with others on. Who had the latest file? Plus, Excel is very linear; you can't easily link a row about a piece of feedback to another row. And its text editing can be frustrating for large blobs of text.
So now I prefer to use task management systems (e.g., Jira) for the purpose of feedback storage. IMO, feedback is best stored in and retrieved from a database, and task management systems are essentially that. You can quickly create an item for a piece of feedback or an entire session, then link that feedback to other feedback items. You can create parent-child relationships, tag items for easier filtering, and it's easy to share these feedback items with others. What's nice is that usually someone else has already set up a task management system, complete with back ups. So it's usually just an issue of having an area of it designated for feedback, especially one where you can control access. Oh, and maybe creating a custom form that's better suited to feedback than a task, if your tool allows that.
I just switched jobs so I haven't suggested we do that yet (since task management systems are really meant for task management), so I'm currently experimenting with Sharepoint. Sharepoint lets you create Custom Lists, and each entry in that list can be defined with a custom form. It's stored on a server (backed up and secured), which allows me to share. I can filter the list view to see just stuff I want. However, Sharepoint doesn't handle cross-linking feedback points well, so we'll see how much I like it.