Get product categories and add them to a custom taxonomy

I'm working on a WooCommerce webshop.

I have divided all my products into categories. Example:

Clothing:
-Tshirts
-Jeans
-Etc.

Likewise, all brands are also divided into categories. Example:
Brands
-Brand1
-Brand2
-Etc.

I want to copy or mirror the category and sub-categories of Brands to a custom taxonomy. So when I have a product in the Brand 1 category, then this product must be automatically entered into a custom taxonoy with the same name: Brand 1

The products in woocommerce are created by an API from a POS system. The reason why Brands is created in categories is that the API from the POS system can only use categories in wordpress and not custom taxonomies or attributes.

It's not optimal to use multiple parent categories as I use an Ajax filter (YITH WooCommerce Ajax Product Filter) to filter my products.

So in general, the category Brands should somehow be mirrored to a custom taxonomy of the same name or something similar.

Is that even possible? And does anyone know how to achieve this? Or does anyone have a better idea?