I can't find a clear example, but it was similar to this but with a more pronounced beeping. Could this possibly be a throwback to the early days of radio or even telegraph communication, or was it simply an interesting sound effect? What did the message mean?
In the example that you gave, the beeping is just a background noise -- it's not actually Morse Code.
But yes, in 1940s-vintage movies, the Morse Code was actually genuine and not just a sound effect. Usually, the Morse Code transmission starts with the letter V repeated several times (the letter V is commonly used as a test character). You can listen for it: it sounds like dit-dit-dit-dah.
This Youtube clip is a good example: https://www.youtube.com/watch?v=vtrPWlZ-Nes
In the sound effect that you gave, it sounds like a stylized version of a Teletype machine. An example is heard here. Teletype machines were a more automated way of sending signals across telegraph and later telephone lines.