It is actually broader than just Germany. Wrapping gowns, as they were called, were popular all across Europe and England. Their source is from the Japanese kimono, as the Dutch were trading with them in the mid-17th century. Likely first brought over for their fabric, but the style became popular as well. You can find examples of them through the 18th century as well. They were worn as an informal coat, acceptable in situations like in the home or in casual company. Their construction location varies, some being made in Europe, others abroad. Importation was big business and even then things were being made in other countries because it was cheaper to do so. And since it is not a fitted/tailored garment, they can be mass-produced easily. They are very closely related to the Banyan, though that has more fitted sleeves and shape. The Banyan comes from India in inspiration and many Wrapping Gowns were made from India cottons. It was all part of the Oriental fashion, so that often got muddled together. You'll find this garment often in portraiture of men lounging in their library, office, or artistic workspace. Very worldly and intellectual.

The garment you're talking about is variously referred to by art historians as a wrapping gown, a morning robe, a nightgown, and a banyan. (There's some disagreement about whether banyan should be reserved for a more fitted garment with set-in sleeves rather than being applied to the loose robes with unshaped sleeves you see in these paintings.) These turn up in many, many portraits of philosophers, scientists, artists, and well-to-do American and European men from the late 17th and 18th centuries; they're thought to be descended from imported Indian robes, and were often made from quite luxurious fabrics like damasked or brocaded silk.

Where women had stays to give them the proper posture, men had the suit. Waistcoats and coats were snugly fitted, pulling the shoulders back; removing these to put on a morning robe allowed more freedom of movement. While they were most properly reserved for private settings - in one's own house, when alone with family - there are references to merchants wearing them in the counting-house, noblemen receiving visitors in them, and gamblers and coffee-drinkers having them on in public.

Being of an expensive fabric, the robe was a sign of personal wealth - but they also seem to have been used in portraiture to indicate intellectual pursuits. "I've freed up my body so that my mind can work," basically. There are frequently scientific apparatuses around the sitter, or they are holding a book or sitting at a writing desk.

A good source is "“Studious Men are Always Painted in Gowns”: Charles Willson Peale's Benjamin Rush and the Question of Banyans in Eighteenth-Century Anglo-American Portraiture"" by Brandon Brame Fortune in Dress, 2002. You can read it on tandfonline.com this week if you navigate through here (click "Explore our Heritage, Museum Studies and Conservation journals" and then "Dress" in the list that comes up.)