Everyday things would be different. You have to account for tails everywhere. Bandaids wouldn’t work with fur. Would shirts be worn, or would natural fur coats be all that is there? Food workers would have full body nets. Claw tips come out of a different area and would make some tasks that use fingertips difficult (typing)…
Assuming we’re talking animal anatomy (and not ‘hands’ on species that normally have paws), all of the jobs and societal roles that require fine manipulation would be limited to a fairly small number of species. Monkeys would likely be very sought after for most of those positions. Rodents would likely be pretty high on the list, too. On the other side of things, canines and felines - despite their popularity - would have the hardest time with things. Never mind things like marine mammals and birds.
Do otters count with marine mammals? Birds are very good at manipulation and some can create some tools. In your description I think that world would have some pretty serious segregation issues.