Garmin, Polar, Apple, and Suunto are some of the brands that offer running watches with music storage. These watches usually have either 4GB or 8GB of storage, which is enough for around 500-1000 songs. To name a few:
Garmin Forerunner 245 Music Apple Watch 7 Garmin Venu 2/2S Coros Vertix 2 Fitbit Versa 3