Does direct damage have to applied to an investigator only or can it be applied to assets?

If it is direct then it must be assigned to your investigator card unless you have an asset in play that has text explicitly allowing it to be assigned direct damage or direct horror.

Exactly as @Assussanni said. There's a third option: if the direct damage is directed at a specific other target, such as on Centuries of Secrets - https://arkhamdb.com/card/05099