Why that word? It could just as easily be a radio stream, or radio feed, terms which are used nowadays for internet-based media (yes, I do know radio very much pre-dates the net. They’re still relevant terms).
Why “station”?
Radio technology did not begin by broadcasting music, it started with Morse Code, & was called “wireless”, as in “wireless telegraphy”.
And it was used for navigation & military purposes.
I would think the ‘station’ refers to the land base from which the broadcast comes from.
When you turn the dial (or press the buttons) you are tuning in the broadcase from that station.
From Etymoline:
‘Antarctic research station’…‘water station’ (I’ve seen that one for long-distance runs)…it’s all the same usage.
Hey, I may be young, but I know that much. 
On preview; thanks, GorillaMan.