Basically it's Eve's very sweet and very refined way of saying "fuck you" to Calix due to the events of Chapter 20 (the flashback during Rosie's 21st birthday).
It's when Eve sat with Calix and a bunch of other dudes during the party and they (Calix, et al.) overwhelmed her by giving two glasses of very strong brandy (possibly tampered).
This puts her in a very bad bind because: 1) according to royal custom, drinking the second glass without drinking the first one is considered very rude so she can't refuse the glasses this way; and 2) when Calix offered Eve to pass one of the glasses to him, this would mean that Eve will be constrained to grant him a favor, which is very underhanded of him.
Anyway, Mikhail takes the glass and drinks it, saving Eve from the bind she was put under.
So yeah, tl;dr read ch 20
Thanks for the reminder and chapter! That was so long ago I didn't realize I saw Calix before.
Please someone, what was the meaning of giving someone a drink here?