Hello.  You state you were 'managing' these blackberries.  Did someone plant them?  Or do you know if they were 'always there'.  If there were always there and presumed to be wild, it is highly likely they are himalayan blackberries.   It is hard for me to determine the type from just the one cutting you attach.  If you know someone planted them, they possibly are a different type that you could propagate  safely

 Below is an extert from an OSU document on blackberries.  The link to the entire document is attached as it has pictures of the types of plants that may help you determine which kind it is.  

'The first species, Rubus armeniacus, has been given a cultivar name, ‘Himalaya’ (Figures 3A, 3B). This extremely vigorous blackberry spreads easily through seeds after passing through birds, and it has become a noxious weed. Its growth habit is similar to semierect types (see page 3). The other introduced species, Rubus laciniatus (Figure 4, page 3), with the cultivar name ‘Evergreen’, was grown commercially for many years before it became endemic. This trailing blackberry has lacy leaves and produces small, sweet fruit in midsummer;'

https://catalog.extension.oregonstate.edu/sites/catalog/files/project/pdf/ec1303.pdf

Most berries, blackberries included, are quite easy to propagate and they do it themselves.  Suckers, or volunteer plants,  occur every year.  We have raspberries and blackberries and we get suckers all the time.  Should you decide to go ahead and try to propagate these cuttings, i would give you some advice:

1)  Try to contain them.  For example, we plant ours into a metal trough to that we can contain suckers.

2) Keep them in check and prune them each year

3) Don't plant them in an area where they can become a nuisance, as  next to a shared fence with a neighbor

If after seeing the pictures in the above document if you believe them to be Himalayan berries, my advice would be not to plant your cuttings, but go to a nursery and get other varieties of berries if you wish to grow them