No, client funds are not company funds. If you run a parking lot and a car gets stolen from the lot you're not liable for replacing the car. You might get that liabilty if your valet wrecked the car, but not in general. Same with deposit boxes, storage lockers, mail packages and so on if you want to get your money back in case of theft you need insurance. Which is what FDIC is for bank accounts. No insurance, then you might not even have a claim against MtGox. First you'd have to take them to court and win to make them liable for damages. And even if you do, well there won't be any money to collect there anyway.
In accounting, generally deposit accounts with customer money are considered liabilities. If a depositor shows up and asks for their money, you are obligated to give it to them. You seem to be confusing legal liability (a "duty of care" to do or not do something) with financial liability (an obligation which must be paid back).
Mt. Gox didn't have storage boxes without knowledge of what was inside them (safe deposit box analogy). They had computerized accounts for each customer, with money in each account. Regardless of whether they were a "bank" they were holding money for other people and that money is a liability in the financial sense.