Animals are permitted in private vehicles but must be properly restrained using appropriate harnesses or equipment, and so the answer is yes, they are allowed on the back seat, but only if they are secured.
As we look at the law, it clearly says that having an animal in the vehicle that could interfere with the driver is illegal and can result in a fine, so a dog sat loose on the front seat for example is not allowed. Carrying animals which are not properly restrained is dangerous to them too. The same applies for cases, cages, and boxes with the animal inside, they too must be properly secured.
Animals are unpredictable, especially in environments they are unsure of or not used to. If loose, they could interfere with the driver´s ability or visibility. In the event of a crash, the animal would suffer considerable injury and, most likely, death.
There are a wide range of security devices available for animals, from boxes and cages, which must also be secured as mentioned, to harnesses which link to the security elements of the vehicle, some adapting the existing seatbelts, others using security points designed for child seats for example. It is not just appropriate to have your dog attempt to wear the seatbelt designed for human passengers, only the appropriate device for them.
Incidentally, if you are found to have an animal loose in the vehicle, there is an 80 euro fine.