Why does this glass of milkshake feel rubbery when I tap it? Every time I get a protein shake on the gym, I am puzzled by the way the glass responds to me tapping it with my fingernail.

If feels and sounds unlike glass, more like a softer elastic material. 
I assume has something to do with the liquid probably having lots of air bubbles in it.
It is strange that if I hit it against my teeth, it feels less painful than if the glass is empty, despite being a lot more heavy. 
 A: Feels is the keyword here.
The glass has a natural frequency of vibration, I am sure you have heard of some high notes cracking a glass.
By pouring in a dense liquid, you have altered the natural frequency but, no offence intended, tapping will give a different sound, and that's explained by above, but I doubt if the actual hardess of the  glass will change in any way at all.  If it really went rubbery, it would not maintain its shape, which it seems to be doing fine.
A: This is interesting! 
Can you try these out next time you go to the gym? 
1) Try to pour water in it instead of milk shake. Does it behave in the same way?
    this would mean that the glass responds in a similar way to all fluid. Then we can think in the direction of acoustics. 
2) Try to pour warm water in it instead of cold shake. Does it behave the same? Maybe the temperature has something to do with this.
3) Can you get a video of this and give a link! I'd love to see such a unique material. I personally have never come across such a material in internet or in real life. :D 
