If it comes to it, I'd just replace it with a chair of similar value, or slightly nicer.

If they're determined to hold back the deposit, the chair's not going to make much difference... at least here in the US you wouldn't be liable for more than the chair costs to replace. Also, several posters said it was a cheap chair, so likely the landlord doesn't care anyway. It could have been bought and left by a previous tenant.

Anyway, the style is similar to a ladderback chair. Maybe there's a more precise name than that, since ladderback chairs usually have flat slats.

I love that blue color though. This is close in appearance but very expensive... https://www.carolinaporchrockers.com/cottage-tall-ladder-back-chair.php

Maybe search for sky blue / haint blue / robin's egg blue ladder back chair...