Stephane,
What I normally do for my stores when they need essentially a product to be displayed in multiple menu types is I use the tags from Joomla and instead of using the product category menu item, I use the product tag menu item.
Basically you need to set one main product category menu item. It can even be hidden. And then creating the brand names as tags. You can tag multiple products by their brand name tags and then have menu items to display each of the brands in addition to the normal category structure. But by using the tag pages, it won't create multiple URLs for the products since they will default back to the main product menu.
You can see it in action on sites like these:
https://www.kjbsecurity.com
https://www.nationwidepowersports.com
https://www.redlinesuperstore.com
Let me know if you have any further questions.
Adam