Not really, with cash you're trusting in the authenticity of a certificate, and from there in the future value and tradability of it. While you have recourse to a list of serial numbers, you don't know who owns what at what time (I'm aware physical cash is a miniscule proportion of economic exchange these days by the way).
With bitcoin I thought that blockchain was a non-editable (for mathematical reasons) ledger of who owns what, and the mining was what you did to create tradeable units. So the blockchain is the record of trades and ownership, and similar technology was used to create more "coins" (because the non-editability means you can make them unique).
I've probably got that wrong at some point, having gleaned bits from several articles that glossed over a lot of technical elements, but if you or someone else here can point out where the error is in a non-patronising manner or provide a link that would be handy (not that I'm likely to be buying in).