I do it all the time. All that sugar acts as a preservative. I use hot water bath canner. I even mix different berries together.
Blackberries do have a lot of seeds!!! I often mix another fruit with the blackberries, like grapes, peaches or apples to reduce the seed to fruit ratio. Or I put them through a sieve to remove as many of the seeds as possible. But we kind of got used to the seeds too.
Homemade fruit syrup is super yummy. But I found that once your open the jar or bottle, you do need to refridgerate it, as it spoils a lot more quickly than the storebought stuff. Probably do to the fact that there are no fakey ingredients in my syrup, only real fruit and sugar