Why does HgCl2 react with HCO3^- but not with CO3^2-?
Anirudh, which book is teaching this stuff? Sorry, I can only find this discussion in 1890s and the latest in my favorite Mellor's Treatise from 1930s.

Coming to 2020: 

After doing a simple acid test for carbonates, a nice safe &amp; sound method to distinguish carbonate vs. bicarbonate salt is 

(a) to react it with a neutral barium chloride solution. If the solution is turbid, carbonate is present, otherwise bicarbonate. Of course the methods fail with a mixture of the two.

(b) Sodium Carbonate-Phenolphthalein Test: phenolphthalein is turned pink by soluble carbonates and but remains colourless by soluble bicarbonates

Your textbook is not completely correct when it states that bicarbonate does not react with $\ce{HgCl2}$. For your reference from A treatise on chemistry, Volume 2
by Henry Enfield Roscoe, Carl Schorlemmer, 1913, pg 699