The referenced link from money.stackexchange.com with the top accepted answer is incorrect. It states:
"Coins are assets because its the actual money. Notes are liabilities because the Federal Reserve is obligated to pay money on these notes."
The Federal reserve is not obligated to pay money on the monetary base it creates and there is no objective financial difference between coins being real money and federal reserve notes being real money. They are both real money for all practical purposes.
Let me take a stab at the initial question.
Yes, Federal Reserve Notes are a liability to the Federal Reserve (as are electronic federal reserve dollars). It doesn't really need to be done like this. It's actually a holdover from when the US was on a partial gold standard. The central bank would hold gold deposits and in turn issue notes (dollar bills) which were redemption slips for those gold deposits. Because the central bank had to redeem dollar bills with gold, this is why it was classified as a liability. The US went off the gold standard twice (once under FDR to the public and later under Nixon for foriegners). When this happened, dollars were no longer true liabilities. But this history is why you see federal reserve notes (and federal reserve deposits which are electronic dollars) still classified as liabilities. And coinage is considered an asset to the Fed which they purchase from the treasury. You will still see some gold as an asset on the Fed balancesheet as a holdover from these times.
The Federal Reserve does not have a monopoly on creating the monetary base. The treasury can also create base money by creating coins and US Notes (which are not Federal Reserve notes).
The treasury does not consider the coin as a liability because it truly is not one (just as the dollar bill in a modern sense is not one to the Fed).
Looking at a balance sheet, you could almost argue that base money is a liability because it appears to be balanced by assets on the balance sheet. But this is not the case. It is an accounting illusion. If the Fed were to withdraw all their "liabilities" there wouldn't be any money in the system (outside of coins and notes of course).
The reference to the obligation on Federal Reserve Notes is an antiquated holdover from the gold standard era. It is quite meaningless now. The Fed will exchange if demanded electronic dollars for paper dollars, vice versa or will buy/sell coinage as required by the banks.
Coinage is not an obligation to the treasury, neither are US notes. The treasury will certainly repair damaged coins, but that is not to say that the coins themselves are true liabilities or obligations.