Heh, Dr. Frank Lang did a little talk on plants of the Pacific Northwest and noted as a joke that getting rid of blackberries in Western Oregon is nigh impossible as there is one single root that runs from Portland to Ashland and grubbing it out is a Sisyphean task.
Blackberries are manageable as a berry vine, don't let the fruit drop so they seed out. Smooth, thornless varieties are available from our local grower's coop.
What you have has been well maintained and kept in trim, enjoy the berries.
As to invasiveness, a Loganberry is pretty impressive as to the sheer growth, it will nearly run you off the property.