Yes, that is a blackberry. As the plants grow and get established, they will get significantly larger. Canes on some varieties can be a couple of inches in diameter at the base if happy. And as your photo depicts, the larger, more mature foliage has five leaflets instead of three.

Definitely a blackberry. Reminds me of the Chesters/TC. Second year growth kind of explodes.

When I first saw it, I thought it was a weed and I broke it off. Well, a couple of weeks later it was back at it again. So I let it grow to see if it would develop any blooms. Glad I didn't touch it anymore.

Thanks for the feedback.