In many-to-many matching markets, various competing stability concepts have been introduced. Most of these stability concepts lack a clear interpretation. This paper argues that the differences between solution concepts reflect different implicit anonymity assumptions. Such anonymity assumptions can best be justified in large markets, modeled in this paper with a continuum of agents. In such large markets, stability is actually a much better behaved solution concept; stability blocks do indeed lead to an improvement for all members of a blocking coalition, unlike in finite markets. It is shown that various differences between solution concepts disappear. In particular stability and weak setwise stability coincide. Moreover, the relationship between anonymity and largeness of the market is made explicit in a natural noncooperative foundation of stability