Signal does have this feature. It's just that their mobile clients do not support it (as a "slave" device), from what I understand this is no different than WhatsApp multi device feature
The traditional "WhatsApp Web" and "Signal desktop" are completely different experiences though.
Though the details differ, WhatsApp's new approach, outlined in this article, is much closer to what Signal has always been doing since it introduced Signal Desktop - which, once linked, functions independently of your phone.