IMO, in this case, "exchange" does not refer to an institution but to the act of swapping financial interest (an exchanging of equities, not the XYZ Equity Exchange).  If your house is worth $500,000 and you owe $400,000 on it, you have $100,000 of equity in your house.  If you also have $10,000 in credit card debt (which the bank owns) you can swap that debt for part-interest in your house.  That is, you can take out a second mortgage on your house for $10,000 and pay off your credit cards.  That would be swapping one debt for another, or an equity exchange.  You still owe the money, but to a different party.

GM might own a factory outright, worth $100 Million. They might trade half their equity in that factory to some bank from which they had previously borrowed $50 Million.  The bank now owns half their factory, and they no longer owe the bank. I expect there would be some language included in the deal explaining how GM might get their factory back.

Edit.  I suppose an equity exchange (swap) might be executed through XYZ Equity Exchange.

I remember some earlier talk that the government bail-out funds should be secured by equity in the companies.  That is, the government, not the banks, would "own" some of the companies' assets.  But I don't think they decided to go that route, because it sounds like socialism.  Anyway, I don't know for sure who will end up owning what.