That's a Russian joke meant for Russians and you talk about "ты" resembling "tea". You need to let go of those cross-language comparisons when studing Russian if you want to achieve good results in the future, just saying.

Anyway:

After 3 shots of cognac a Frenchman switches to mineral water instead, while a Russian starts to adress people on a first name basis.

It's hard to give an accurate translation of the last part though. Переходить на "ты" is a Russian thing. You might know that "вы" can be both a regular plural "you" and a polite singular "you". If you don't know a person well enough, you adress them with "вы" as a sign of respect to them. When you're friendly enough with someone you just adress them with a regular "ты".