It is part of M0. It's more "money" than deposits (M1) or savings accounts (M2). In a discussion about money supply it definitely counts as money. And they have it. Yes, they need to keep enough reserves but apart from that it's their money to use as they please (lend, invest, spend, pay dividends...).

The deposit is their liability (obviously they can't spend it). The dollar is their asset. When you deposit the money into a bank that dollar is not yours anymore, the bank is not keeping it in custody for you.