One db of products with multiple domains

Hi,

If you split your products in different category trees, and then set the different main categories as main category of the menu item allowing the customer to access the shop, and then use a Joomla extension to hide/display menu items based on the current domain name, you should be able to achieve it within the same website. For example, this extension should allow you to do that:
extensions.joomla.org/extension/chameleon/

Otherwise, JMS should also be a solution here where you would manager your shop through the master website and have one slave website for each domain name.

However, for both solutions, I would recommend to inquire the developer of the extension about your situation as I'm not a user of these extensions and just know things from looking and reading around. They will be able to talking about the capabilities of their extensions much better than me.