No, not all the stainless steel used on milk pitchers is the same. In some cases the stainless steel is not even suitable for food contact. Some of the cheaper milk pitchers are made from 202 grade stainless steel, which is not suitable for food applications.

Yes you can use a painted milk pitcher, but never one which has a coating on the inside. Seen the paint is likely not suitable for direct food contact.

Yes you can put stainless steel milk pitchers in the dishwasher. For the painted stainless steel milk pitchers it is not adviced to put them in the dishwasher. Those need to be cleaned manually.