I think this is related to traders jargon. When a dealer quotes the price of a spread between two securities (such as a risk reversal) as "10 cents your choice" or "ten cents around" it means that the bid-ask midpoint is zero and it will cost you 0.10 USD to enter a position long the first security/short the second, and also 0.10 to short the first/long the second. Obviously you have to tell the dealer which one you want to do. In the more normal situation one security is worth more than the other and then the spread might be quoted as "forty twenty for the call" meaning you pay 40 cents to buy the call and sell the put, or receive twenty for the reverse.
By extension (and I have not heard the term before) it would seem the "near choice" would mean close to zero, but maybe not exactly zero. So yes, it would seem to mean that ATM puts and calls are approximately the same price.