Autoscaling groups start multiple EC2 instances. Each EBS volume can only be attached to a single EC2 instance. Therefore you shouldn't do this, and it's probably not possible. If you have multiple volumes and multiple instances you can probably implement some complex logic to make it work, but you shouldn't - simple architectures tend to be best.
What you can do is run an EC2 instance sharing data over NFS that the ASG accesses, or you can use AWS Elastic File System, but beware latency is relatively high.
If you expand on your requirements and use case you may get more useful answers.
Based on the additional requirements, it's probably possible. It's not a standard requirement so I don't think there's any explicit support for this.
I would probably look at writing some kind of startup script which associated the volume with the EC2 instance then mounted it. There's an answer here which could be of some use. It's probably possible with some kind of setup with SNS and Lambda, but the EC2 instance would likely still need something to run to map the volume.
Someone else may be able to give you more information, I've never had to do anything like this.