Can Azure storage account (files) be mounted on multiple servers?

I am using storage accounts (files) as a mount point on a Linux server. It's being used to storage media files.

Can the same volume be mounted on another Linux server? Will this cause issues?

The goal is to have multiple servers sharing the media files, for read and write purposes.

If you're referring specifically to Azure File Storage, which is an SMB 3.0 share that sits atop Azure Storage, then yes - you may mount this file storage to multiple VMs. Since you mentioned Linux: There's currently a limitation where the VM must be in the same region as the file share (see this article for more details).

If you're referring to Azure blobs directly (specifically, a vhd mounted as a disk), then no - a vhd may only be mounted to a single VM (and there's a blob lease preventing any other VM's from also mounting that vhd as a disk).