That's BS! Binance (as far as I know) doesn't require users to withdraw only to their own addresses. That would for instance make it impossible to transfer funds directly to another exchange. It's quite common practice for users to make a payment directly from an exchange or to another person. And on top of that, even if you owned the address, Binance doesn't require you to keep having access to the address you used 19 months ago.
They're now just trying to grab funds from customers to limit their own losses. I assume they'll get lawsuits over this.
Excuse me, maybe I put it wrong because I use a translator. I meant that binance believes that I have access to the address to which I once displayed DOGE. Regardless of where the transfer was made. Whether it's an exchanger, another user, another exchange, your wallet or not yours, and so on, there are many options.