I'm starting out building my business and plan on only ever having one product. I'm wanting input on the branding of my business and of my product. I see two ways forward. One I brand my business MyBusiness and my product CrazyProduct as an example. The second way is to use the same name to brand the business and the product.
Google, Microsoft, and Blizzard follow the first example, and Twitter, Facebook and ebay follow the second.
The big differences I see is that those following the first example offer many products, and those following the second only one.
So I am inclined to choose the second method, that is name my business the same as my one product.
If you "plan on only ever having one product", then you're not building a business that has a product, you're building a business around a product. Obviously, there will have to be some unknown entity ("CrazyProduct Co LLC") that produces this product, but that's a footnote on a legal document and should remain unknown.
I can say from experience with Inedo Media, brand building is critical and it's very challenging. Building two brands is not only double the effort, but it introduces confusion, which chips away at the chances of success.
Stick with a single brand.
I would have a separate name for the company and the product, this keeps your options open for expansion down the track. As far as perceptions to clients/users, there is no need for them to even know that your company is a different name behind the scenes, you could have your marketing website setup for "CrazyProduct" with no mention of your company name on it. You could then also have a basic placeholder website for MyBusiness if you wanted it, but it wouldn't be necessary unless you plan to tell people about it.
The advantage to this approach is that any infrastructure you setup to get going can be done under the MyBusiness name, and if you get half way through CrazyProduct and decide a change of plans is needed, your infrastructure is still all associated with your business (by infrastructure I mean your email address, possibly any legal asset ownership if you are taking it that far, etc.)
This also leaves open options to do some additional consulting or freelance work on the side under your business name.