Can I assign a keyword to multiple products?

No, that is not possible. Otherwise there would be the possibility that you bid against yourself and thus artificially increase the price.

Google states:
"Because we never run more than one ad from a given advertiser at a time, identical keywords can have a negative impact on the performance of your ads. If you use the same keywords for two ads, they're competing with each other, not with other advertisers."

If the keyword is already assigned to a product, category or brand and another attempt is made to add exactly this keyword to another element, Shopstory will inform you of this. You can then choose whether you want to leave the keyword with the previous shop element or whether it should be moved to the new element.

In principle, we recommend to use a keyword where it fits better. It fits better where the customer finds the right product, ideally with matching SEO text.