Why have an inflatable goatskin that you're inhaling from for swimming on the surface? Considering the fish next to him it seems more likely the artist left out the boring aspects of holding rocks or whatever to reduce buoyancy.
Artists don't always use realistic perspectives, especially not that long ago. The swimmer isn't inhaling, he's blowing into the goatskin to keep it inflated. It's just like a modern inflatable life vest which has a manual inflation tube to use if the automatic inflator fails. There are no rocks (although ancient free divers sometimes did hold rocks to dive down faster).
Who says he's breathing from it? More likely he's breathing into it (like bagpipes) to keep it inflated. It's not likely to be perfectly airtight.
Did you also consider the horse depicted below the fish?