In an update made to the web version of WhatsApp, it seems that the web client now supports picture-in-picture mode. This is similar to the mobile version of WhatsApp which allows videos to be played in a separate floating video player. This lets users continue to chat with their contacts, move to a different chat, all the while the video is still playing.
However it seems that the web version might actually be better and more useful because it means that users will be able to play the video while chatting with their friends and continue to use other applications at the same time. So far only a handful of video platforms are supported which includes YouTube, Instagram, Facebook, and Streamable.
In order to take advantage of this new feature, you will need to access WhatsApp via the web. We know that there is a desktop client but for now this feature is limited to the web version, but perhaps it will arrive to its desktop counterpart at a later date.