It looks like it will depend on the beach, some beaches have restrictions on where you can launch, some don't and some do not allow kayaks or paddles sports.
It would probably be easiest to ask the park department which beaches allow kayaks or you could go through the list and see if a particular beach allows it.
This Cabo da Roca guide offers three options: a badger, an Egyptian mongoose, or a weasel.
Badger is quite big and, according to East Surrey Badger Protection Society, "The front paws of an adult badger range from about 4.5 cm to 6.5 cm across for a large boar". The print is also different - see, for example, here.
As for mongoose - I couldn't find any ...