Jalapeño cheese is cheese — any cheese — with jalapeño peppers in it.
There are a few commercial brands. Some are a semi-firm cheese such as Monterey Jack with jalapeno pieces in it. Others seem to be made from the mild, white semi-soft cheese called Queso Quesadilla. Some of these soft cheeses can be heated and used as dips.
Most commercial jalapeno cheeses seem to be aimed at the U.S. market, as most Mexican soft cheeses seem designed to be mild, with the spiciness in the food coming from elsewhere. Mexicans would typically just use the Queso Quesadilla cheese, and add the chopped jalapeño separately to what they were making.
There’s nothing to prevent you from getting a tub of cream cheese, stirring in a chopped up fresh or pickled jalapeño pepper, and calling it jalapeño cheese, because it would be.