American Express is both an issuer of credit products and the payment network on which the cards it issues run on.
American Express also licenses other issuers of credit products to run on the American Express payment network.
An example of such a card would be the Wells Fargo Propel card.
The Wells Fargo Propel card is closed for new applications, but it was one the better known non-American Express issued cards running on the American Express payment network.
A less obvious example would be the Macy's American Express card.
The Macy's American Express card is issued by Citibank.
For cards that run on the American Express payment network but that are not issued by American Express, rewards and sign up bonuses will usually be in units that the issuer defines and provides, not American Express Membership Rewards.
Towards the beginning of the Covid19 pandemic, on Twitter American Express publicly stated a maximum of 4 credit card products.
This has since been relaxed to the pre-pandemic soft-cap of 5 credit cards.
There is also believed to be a maximum of 10 charge card products per user.
For any one person it is possible that the maximum accounts allowed may be higher or lower than those values.
Cards that run on the American Express payment network that were not issued by American Express do not count against those maximums... they are just other accounts in your credit reports just like a Discover, Mastercard, Visa, or any other account.