The payment processor doesn’t belong to any of the credit card companies if they are tied to any institution they will be part of the acquiring bank but these days most of them aren’t, all they do is facilitate the transaction and there could be more than one in the chain.
The issuing bank as in the institution that issued the card charges a fee, as well as the acquiring bank the bank that actually receives the payment also charges a fee.
There are fees that are levied on multiple parties across the chain by multiple parties however at the end these tend to be mostly be then baked into the fees that are levied on the merchant.
Each of them charges a fee, and can be charged a fee. A credit card transaction is a multiparty transaction with a lot of small fees.
So let's say that you swipe your card at a retailer, perhaps an IKEA.
IKEA has a card payments processor (ie: the vendor that the little terminal you sign on / stick the card into is from). This processor deals with a network (usually Visa or MasterCard, although there are smaller vendors such as Amex, Discover, etc) who ultimately connects the payment processors to the banks.
Now, for each transaction, your bank charges a fee to the network (the interchange fee, limited in the EU to like 0.3% I think) as a percentage of the total transaction. The network then charges some fees to the payment processor, which includes the fee that the bank charged. Finally the payment processor adds in their own fees and charges some unified fee schedule to the merchant (in our case, IKEA).