I expect it requires replacing the whole sash if the window maker does not sell the part separately; if it's typical vinyl sash welded at the corners there probably isn't a separate replaceable part. Start with contacting them to see if they do, you're not going to find that at the hardware store. If you can't get the replacement part or a replacement sash you might need to replace the whole window.
That would be the amount to deduct from the security deposit, as that would be the way to get it back to the way it was before dog damage.
I can envision a practical repair of making a wood piece with a slit to be glued (and perhaps screwed though a couple of holes drilled in the intact parts) in place over the plastic projection, or two strips of wood (one above, one below) glued and screwed to it (the slit approach hides the damaged edge better, though.) That will look odd if it's the only window with that detail, but done well it should not look bad. Might be worth doing all the windows in this apartment so they look the same.
I suppose that could be done using plastic trim "lumber" if you wanted it to be all plastic. I'd consider properly done wood an upgrade, but tastes vary.