Answer: It's a type of pacifier called a sugar tit:
Pacifiers were . . . a substitute for the softer sugar tits, sugar-teats or sugar-rags which had been in use in 19th century America. A writer in 1873 described a "sugar-teat" made from "a small piece of old linen" with a "spoonful of rather sandy sugar in the centre of it", "gathered ... up into a little ball" with a thread tied tightly around it. Rags with foodstuffs tied inside were also given to babies in many parts of Northern Europe and elsewhere. In some places a lump of meat or fat was tied in cloth, and sometimes the rag was moistened with brandy. German-speaking areas might use Lutschbeutel: cloth wrapped round sweetened bread, or maybe poppy-seeds. A Madonna and child painted by Dürer in 1506 shows one of these tied-cloth "pacifiers" in the baby's hand.