Yes, coffee beer contains caffeine… but only in very low amounts. Most brewers use a much higher ratio of beer to coffee (because the beverage is labeled as a beer, first and foremost). A typical ratio might be a 31-gallon barrel of beer to a cold brew that's extracted from just one pound of coffee beans.
Coffee beer is made by adding dry coffee grounds into grain during fermentation or by mixing cold-brew coffee directly with beer. Coffee beer often referred to as coffee stout, is mainly just beer with subtle hints of coffee flavor.