what prevents someone from coming forward with the same trust name and attempting to lay claim to such assets
The fact that they won't have a notarized document with your signature on it establishing the trust, whereas your trustee will.
Running a quick check on things owned by trusts, the names in this state for revocable living trusts tend to look like the following:
Firstname Middlename Lastname Revocable Trust Dated mm/dd/yyyy
Firstname Mi Lastname Revocable Trust Agreement Dated mm/dd/yyyy
Firstname Mi Lastname Revocable Trust Under Agreement Dated mm/dd/yyyy
Lastname Family ...