Steam Family Sharing Is Not Supported
Due to Last Epoch being available for purchase outside of Steam, registering a Last Epoch account and linking it to your Steam account is required to play the game.
Unfortunately, this is not possible for accounts using Valve's Steam Family Sharing feature. As noted on their official page regarding the feature;
Can all Steam games be shared with friends and family?
No, due to technical limitations, some Steam games may be unavailable for sharing. For example, titles that require an additional third-party key, account, or subscription in order to play cannot be shared between accounts.
For those using Steam Family Sharing, we recommend purchasing Last Epoch outside of Steam and using our standalone client.