I think that having your hand in that position while having your picture made was popular back then. ….. For men. But not for women. That would have been naughty.
You will see that a lot. The hand inside the coat in photos is supposed to be an indication of leadership during the 18th and 19th centuries. -- Wikipedia
Apparently, that has been done since 300 BC but not continuously. Most of Napoleon's portraits were painted with his hand inside his waistcoat, then, in the time of photography, it became widely popular for a century or so.
Well, whatever or how thin, Lincoln's coat is a little tight. E!ither needed to lose a little more weight or be fitted for a larger coat. That one is straining.
I got a giggle out of one sentence in your link, Ken: "Francois Nivelon's, A Book Of Genteel Behavior of 1738 noted the hand-inside-vest pose denoted 'manly boldness tempered with modesty.' " Wonder if that will ever come back in style?
Robert Frost with his daughter Leslie, and her two daughters, during a game of pick up sticks (1945). How could you play pick up sticks on a rug?
With the light stand and cords around, it looks like a posed picture. The dog looks like he is part of the rug.
Could also have been something quite trivial like the two pairs of glasses and case(s) that Lincoln seemed to always carry with him because he needed them but which he was not wearing when the photo was being taken. Alternatively, his watch fob and/or silver pocket knife and wallet. https://artsandculture.google.com/exhibit/dwIS9Mh5nGISLg