How to prevent WhatsApp (Web) from losing received media?

The described behavior is particular to the use of WhatsApp Web (or, maybe, also the WhatsApp app on the computer, I did not try).

I typically use WhatsApp from the web browser on the computer (WhatsApp Web), not from the phone. When I click on a media in WhatsApp Web, the media is shown, and I assumed that it would be synchronized with the phone (downloaded to the phone, saved there, and then shown on the computer). No. It is not downloaded to the phone, only to the computer. What is more, it is not saved on the computer but is only shown directly in the app.

Consequently, as time passes and the original media is no more available for download, there is no copy saved on the computer, and there was never any copy saved on the phone.

In order to have the media downloaded to the phone, you need to open the WhatsApp physically on the phone and tap the media. Presumably, after that the media would be saved to the phone and will not disappear anymore, as the other answer indicates.