What does it mean when we talk about "same basket" exactly for both GDP and CPI?

Interesting question. I am not entirely sure of the answer below but here's an attempt:

"Assume that the basket used for calculating the CPI is the same as the composition of 
 GDP. Will CPI in a given year then equal the GDP deflator? If so why, if not why not?"

Answer should be NO.

So CPI comprises of basket that participants consume, and * private consumption* is only a part of GDP. From what I understand CPI basket being same as composition of GDP means is that in the consumption expenditure $(C)$ part of GDP $(Y)$ has same composition as the weights in CPI basket, i.e., if 30% of private consumption expenditure, say, comes from spending on food items then the weight of food in CPI basket will be exactly 30%.

So as such CPI can be used as a perfect deflator for deflating $C$ but not $Y$. The GDP deflator would have to additionally take account of private investment, and government consumption and investment which may or may not have same composition as that of CPI basket.

Same basket means same goods and services at factor cost or market price. Weighting method could be different.