What is the ruling on naming organizations and products with foreign names, and does the ruling differ depending on whether it is in an Islamic or non-Islamic country, or whether its ownership belongs to a Muslim alone or to a Muslim and a non-Muslim partner?

There is nothing in the Shariah that prohibits naming a region or a product with any name, and the default rule for things is permissibility unless there is a prohibition. Therefore, there is no harm in naming a region or a product with a non-Arabic name, unless it is one of the names forbidden by Shariah, such as the names of idols, or names that imply servitude to other than Allah. There is no difference in this regard whether it is in an Islamic country, owned by a Muslim, or a company jointly owned by a Muslim and a non-Muslim.