Note that the Desktop mode is available for Windows and Mac OS while the Web works on any internet browser but requires linking accounts from mobile phones. WhatsApp said with the introduction of the new feature, users will have to authenticate their phones while linking/logging into their Accounts to the Web / Desktop.
From 2019 onwards we have seen quite a few evidence of fingerprint authentication on Android Beta. Notably, after rounds of Beta testing, the feature eventually made its way to the final version. Shortly before this, iOS users also got the authentication but it was Face Unlock because of the absence of fingerprint on newer flagships. It seems WhatsApp is also focusing on PC as it recently got support for Audio/Video calls. In simple words, this new update for Web / Desktop adds an extra layer of security.
WhatsApp pointed out that the authentication is taken care of by respective APIs and it doesn’t have access to the biometric data. Note that this feature is not currently available for everyone. It will start rolling out in Androids that have fingerprint capability and / iOS above 14+.
Once the feature gets rolled, by default, you have to authenticate the linking. On contrary, if you wish to delink, you have to disable the fingerprint / face unlock on your phone too. According to experts, the next big update could possibly be multi-device support.