I understand what you are looking for and I can not think of a Chicago equivalent. Others have done a better job of mentioning stores close to what you are looking for than I could have.
You specifically mentioned their babka - it is awesome. I am there at least once a month and buy one virtually every time.
As good as the store is, I find their website very annoying: prices are much higher online than in the store. Babka and their coffee are 2 examples. You get a signup offer and occasional offers online but they are not that good. I unsubscribed several years ago.
As you mention, there is one Zabar but as I am sure you know, there is Elizabar's on the Upper East Side.
http://www.elizabar.com/