Even if they are partially at fault, the company tends to have more power to fix security problems than the customer does. That's why we tend to put the onus on the company to fix these issues. It's not really fair to put it on either one for something criminals did, but at least the company has more power to control things.

In the case of credit cards, the US industry has implemented PCI compliance to force a level of security on all the individual companies. Now, I happen to think PCI is a flawed approach. Payment gateways in most other countries work something like PayPal or Google Wallet, where only the processing company ever sees payment data. The merchant only sees that the payment is verified and has the correct amount. However, US internet sites evolved where each individual merchant has to hold on to credit card data, and that necessitates PCI. Fortunately, PCI compliance is such a PITA that many companies are turning to payment gateways like everywhere else in the world.

In the case of 23andme, they had a few broken passwords that then affected half their customer base through the relationship feature. Aside from dropping relationships, they also could have used MFA methods. My Steam account uses MFA, and it's far less important than my DNA information.