Yes—use a Photoshop Action together with File > Automate > Batch, but don’t try to record a normal copy/paste from an open mask file. That approach tends to fail in actions.

Instead, save the mask as a selection first, then use that selection inside the action:

1. Before batching, open the mask source and use Save Selection.
2. Create an action that does something like:
   • Open image (or let Batch supply the files)
   • Load Selection
   • Add Mask
   • Save
   • Close
3. Run the action with File > Automate > Batch on the folder of images.

The key is that Photoshop actions handle save/load selection much more reliably than trying to copy and paste mask pixels from another open document. Once the saved selection is loaded onto each image, adding the mask applies that same shape to every file.