Having assets under management doesn't mean you have that money. You don't own it, you are just taking care of it for somebody. When describing a company as an $X billion company, conventionally this is referring to the market cap. You could use it to describe other things they possess if you wanted to, but assets they manage will never be something they possess.
Language is a communication tool. If you misuse language you will be badly understood. The solution is to use the correct word for what you mean, not to accuse others of sophistry.
