Difference between Fleischkäse, Mortadella, and Parizer
Your instinct is correct, all your favourites fall into one category of saussage: Brühwurst.

This means that they follow the same principle when made, which is


Mincing the raw meat and fat with ice or ice water, salt, often curing salt and various spices until a smooth paste forms. This is what you refer to as "pinkish mass".
Shaping the saussages (or loaf in case of Fleischkäse) and sometimes smoking them.
Heating the saussage until the core reaches around 68 C / 154 F, either by scalding in hot water, baking (for Fleischkäse) or roasting. Sometimes this step is left for the customer, e.g. in the case of Fleischkäse that can be baked in the home oven, raw Bratwurst for grilling or Weisswurst.


The differences are in the spices and additional ingredients (pistachios, mushrooms, cubed fat for Mortadella...) details in preparation (coarser chopping or smoother mass, smoking or no smoking, shape) and the percentages of meat / fat / water (typically around 50%/25%/25%).