When the signature field with the tenant's name is blank, it is because the individual has chosen not to have the name displayed on certificates when signing with MitID.
This applies particularly to those who have a secret address, secret identity, or are protected against advertising. When signing with NemID, the name is replaced with the word 'pseudonym', while with MitID, the field is left blank.
The name in the rental contract builder will therefore be the only indication of the tenant's real name. However, the signature is no less valid, as it is linked to the tenant's MitID registration. This means that signature is associated with a CPR or CVR number regardless of whether the name appears or not.
You can read more about MitID here.