I'm going to state something you probably already know
Agile is about the team - not about the products.
It feels like a very obvious truth to me, but I didn't realize it until recently. It was one of those things that as soon as I said it, I realized that it was extremely true. I hadn't heard it before (or it didn't stick - I'm pretty dumb, after all).
Why haven't I heard this before? Why does this even matter? This is probably due to, at least in part, always working on small teams. On the rare occasions where I've been on a larger team (6-7 developers), we had several products. Usually to the point where each developer worked on a product by themselves, or mostly by themselves.
Now I'm going to talk about more things you probably already know!
How things typically seem to go:
Traditional (typically rooted in an attempt to do Waterfall) organization in regards to software development seems to be centered around the products. There's a request for some new software. It gets added to a Product Manager's "list of products I'm PM for". The PM makes a request to the development group. The development group allocates some resources to the effort. This is done by taking one or more developers away from something else they're doing - or worse, they're asked to "split time" between multiple products. Each way has their own major problems.
If a developer is asked to split their time, they now have conflicting priorities. They can ask whoever represents the Product group which product feature is the most important across multiple products, and will get different answers. Everyone will always want whichever product they're closest too (or is the squeakiest wheel) done first. Getting a single clear direction, that doesn't change on a daily (or worse, hourly!) basis will be neigh-impossible.
If a developer is made dedicated on this new product, while at least they won't have conflicting priorities, other issues arise. That developer will likely be working in a silo. No ability to do peer reviews (other than simple adherence to coding standards possibly). Higher stress due to not being able to take sick time. Developing in a black box. Harder to bounce ideas off of peers, since no one else will know the domain. Chances are if someone else has to join in on the product later, that person will be very lost, and will likely end up needing either a lot of time learn the existing codebase, or re-writing the product.
In both cases, any existing products are damaged, as their ability to do planning will be interrupted. If they're attempting to do Scrum, for example, now their velocity will be incorrect, and will have to be recalculated. Again, if things are ran by product, and not by team, and especially if a developer is spread across products, getting any consistency will be extremely difficult, if not impossible.
How organizing around the team fixes things:
When you organize around the team, in a proper Agile environment, one of the first things you'll get is a single Product Owner. Without getting into the full definition of the PO role in Agile terms, the key part that is important in relation to this post is the fact that the PO will set the priority of work for the team. This gives a single person the responsibility of knowing what's important to each product the team supports and the relative priority among the products. This is critical.
The next thing you'll get is predictability. With the developer/QA resources (boy do I hate using that word, but stick with me) being constant, the amount of work the team can get done will be consistent - regardless of the product.
You'll also avoid silos this way. Since a team grooms stories together, everyone will be knowledgeable about the products. This gives people freedom to take time off work as needed, without being "the one guy" that knows how something works. The Company typically likes this aspect when phrased as "If we do this, then if [So-and-so] gets hit by a bus, the product isn't doomed!". If you're practicing some good policies like peer reviews, that will also help spread knowledge about the products throughout the team, as well as let everyone feel that they know all the products on that team.
Avoiding possible pitfalls:
What if it gets to where you have lots of products (typically, several smaller ones and a larger one or two)? Then you have another problem - but it's not solved by organizing around the product. Not that I'd even really want to call that a problem. If one product consumes 80% of the team's time, it may be viable to split the team at that point, so that a larger portion is dedicated to that product, and a smaller team is formed to handle the other products. Be prepared to keep the teams consistent after that point though, and don't start "borrowing" resources from other teams, or you'll end up right where you started.
Splitting the team also means getting another PO - one for each team. It might, in theory, be possible to have a single PO over multiple teams, but more than likely it'll end up with that person not being able to focus on each team as they should.
Many businesses want "on demand" time spent on their product. They want that product right then and there, and later on, interest in that product might slow down (especially once it's been launched, and initial revenue gained). That leads them to want to shuffle development around as needed. Agile is great at providing visibility in the effect these decisions have. By switching the focus from the product to the team, it allows businesses to see how it actually hurts other products, instead of creating the illusion that development is an unlimited resource. It's up to the PO to communicate back to the Powers That Be that the new product can be delivered, but that existing products will have their timelines affected. It's going to happen anyways - but when focusing on the product, there's the false sense that other products are unaffected.
Often the most important thing when selling software (or selling anything, honestly), is managing expectations. By organizing around the team, you will get consistent, predictable results for your clients. Your developers will be able to focus on the work laid out for them, instead of trying to juggle priorities and emails. If you can deliver what you say, your clients will be happier. You'll have happier developers this way too, which lead to less turnover - which will again help with more predictability, feeding back into happy clients.
There are other things I feel are beneficial as well, but this post is long enough! Suffice it to stay, focusing on the team helps solve many problems, and will prevent other issues from arising. Again, this is based on my experience, which is limited to tiny teams, or a larger team with more products than they can handle.