Are there Chinese spoken language games? (like Ubbi Dubbi)
To my knowledge, I used to play a game called 成语接龙（
Idiom Solitaire， perhaps), which I think it may be similar to the spoken game that you just mentioned.
In the 成语接龙, you are required to say an idiom, and your frind have to say anthor idiom starting with a Chinese character whhose pronunciation is similar to the final Chinese character's pronunciation of your idiom.
Example: