No account, no email address, no document upload, no cookie that follows you. Here is what that does and does not buy.
There is no registration on TornadoSol because there is nothing to register. Your note is generated in your browser, the deposit is a transaction you sign yourself, and the withdrawal is a proof your browser builds locally. At no point does the service need to know who you are, and it has no facility for storing it if you told it.
Honesty is more useful here than a marketing claim. Your deposit transaction is on a public blockchain and so is your withdrawal; the point is that they cannot be joined, not that they are invisible. This site is served over the internet, so the server sees request IPs like any server does, and the page loads Google Analytics, which means Google receives visitor IPs and visit times — and visit times correlate with deposit times. If that matters to your threat model, use Tor or a VPN. A privacy tool that quietly lied about this would not be worth using.
Deposits are held briefly before entering the pool, so the operator can decline and return funds — which also means the operator sees the incoming transfer. Once a deposit is in the pool, the zero-knowledge withdrawal hides which note is which from everyone, including them. The residual risk is a thin pool, where most deposits belong to the operator; that is a numbers problem, printed on the pool picker rather than buried.