So I essentially want to see in Salesforce which products a given account has. I know that you can add products to Salesforce, and associate them with an Opportunity, but I'm not sure how to add them to an account. Perhaps a note would be better?
You're wanting to use Assets. You can create some automation to have Won Opportunities move the Opportunity Line Items over to the Account as Assets (processbuilder & flow would do this if you don't want to use code). You can add custom fields and layouts to the Assets so that it only is showing you want matters.
If you really want to start adding Products from the Account itself you may have to go with another Junction Object called AccountProduct.
AccountProduct will have lookup to both Account and the Product.Depending on the functionality needed you may have to do lot of customization for your application .