They are a pension fund; they literally had/have US$125 billion dollars under management. What exactly is being stretched here? I can't for the life of me think of something that qualifies more for being a 125b company than actually having 125b in assets.

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.

Companies are described by revenue. UniSuper made $110 million recently. It deceptive to use the assets managed as the size since it makes it look like a much larger company. NVIDIA has revenue of $130 billion. $125 billion revenue would make it the largest company in Australia by a good amount.