Does a phono cartridge have a break-in period??

The short answer is "yes". Regardless of what you think about break-in of electronics/wires, etc, there is no question that the elastic characteristics of suspension components will change over time. So will the "shape" of the stylus. Unfortunately, that also means that they will eventually require replacement. Luckily for you, with an MM, you can get a new stylus/cantilever/suspension to plug in for relatively modest $.

"My question is: do cartridges have a burn-in period after which the sounds is fuller or better?? If so, what exactly is breaking in?? stylus, cartridge coilings??"

Of all the components in an audio system, the phono cart changes the most in sound from break in. Its not even close. I've had cartridges that sounded defective when they were new. For the first 10 or 20 hours, you can usually hear a difference from record to record.

As far as what's breaking in, its a mechanical device like a speaker.

I find, for most cartridges, 40-50 hours is about the average for break-in. Some less, some more.