is it possible to render a picklist field in a field editor as multiPicklist?

A lookup record can only contain a relationship to one other record - so you won’t be able to select multiple Products for that one case. If your users will only ever need to select 2 or 3 records, then you could create another couple of lookup fields on the Case object.

But if your users might want to select ‘many’ related products, then you will need to look at building a Junction object between Case and Product - check out the documentation here on a ‘many to many’ relationship. This would mean that rather than having a ‘field’ called Property, you want to create a new ‘Custom Object’ called Property, with a master-detail lookup to Case, and another Master-detail to Product.