what do you call the mixture of melted ice and left over alcohol in
the bottom of a cocktail shaker, that people sometimes drink

Yummy!!!!!!!

During the time of royalty in Germany, the servants cleaning up after
feasts would collect the leftover drinks in a hogshead or similar
container, split it up amongst themselves, and enjoy it then or take
it home (usually for further fermentation). This was called the "kalte
ende", or cold end.

Later, a translation error changed this to "kalte ente" or Cold Duck.

I like the phrase cold end for the leftovers. Pretty descriptive, no?

Hmm, despite decades in Germany, I never heard (or found here, now)
that explanation.  These days Kalte Ente is a bottle of white wine, a
bottle of bubbly and lemon (or non-alcoholic: apple juice, soda water
and lemon).
Anyway, in my family, there was never anything left over in the
cocktail shaker, it was served as seconds.

Normally  - less attractively expressed than Steph has -  anything
left in a glass, bottle, barrel or other drinks container is called
the dregs.

Thanks for all that, but there is a specific term for the mixture of
residual alcohol and melted ice from the bottom of a cocktail shaker

"Slush"?