Dog footprints are oily, like hand grime ... try mineral spirits.

Buff and re-coat and keep the dog's nails trimmed. Or, do a sand and refinish to a lighter color that shows the issues less.

That's not a great picture but that does look like scratches, not footprints. It almost looks like the poly is de-bonding from the stain.
If you're going to consider a recoat you probably should try a test area first.
Besides clipping the dog's nails I've seen the suggestion that you also trim the fur that sticks out between their toe pads so they can get better traction without extending their claws.

I know that this post was 7 years ago... but I’m hoping that there homeowner came up with a solution as I am having the same issue. the problem is not scratches, it’s like a oily film that show every dog print and smudge. Please let me know if any solutions were found. I have now refinished the floor twice to try to fix the issue and it’s still not resolved. :/.

@elizabader It is best to start your own thread with photos. Many of us won't 'reengage' a very old thread if we don't have to.

Ok. Thanks :).