Dealer-installed alarm sounds when passenger door, trunk opened

My father recently purchased a 2012 Prius 3. To reduce his insurance premiums, he purchased an alarm system, which the dealer installed for him a few days after he first took the car off the lot. He doesn't know which alarm system this is. Now, when unlocking or locking the car from the driver's door handle sensor or via the remote, it both chirps (not the horn) and briefly sounds the horn twice or once, depending on whether you are unlocking or locking. This isn't too much of a problem, but it might help you identify whether this is a proper Toyota alarm system.
The problem is that when the car is unlocked from the trunk or the passenger door (using the door handle sensor with the key in pocket), this does not disarm the alarm, and the alarm will sound when the door is opened. The alarm is only disarmed when using the remote's unlock button or using the Smart Key sensor at the driver's door. One of the major differences between the 3 and 2 trims is that the 3 has Smart Key capability on the front doors and the trunk, rather than just the driver's door -- but the alarm has negated this benefit, as opening the trunk or passenger door without first unlocking from the driver's door or remote causes the alarm to sound.

Is this normal for a Toyota-installed alarm?

 

I suspect it's more about the contents of the car than the actual car -- the Prius already has electronic ignition and all that jazz, and is thus hard to steal, but without an alarm, someone could break a window without being noticed. I do suspect that it doesn't really do much to prevent theft, though...