Spreading money between different brokerages and fund companies does provide institutional diversification to mitigate counterparty risk. Most people don't really understand the levels of ownership between you and your stock/bond/asset when you buy into a fund.
When you buy a fund like SPY, you actually have at least two companies between you and your stock. Your brokerage (Fidelity, Vanguard, etc.) and the financial firm running the fund/ETF (State Street in the SPY example). Some funds like gold ETFs and REITs are even more complex with multiple layers and parties involved. Then there's the layers of securities lending that goes on inside many funds, where "your" shares are often lent out to companies shorting the same stock (they virtually all do this -- even Vanguard). You really need to read the prospectus! The large companies have good records but are still run by human beings. If you had all your money in SPDR funds you'd be in trouble if State Street ever had issues as a company. Same with iShares and BlackRock. Buying a "house brand" Fidelity or Vanguard fund removes one extra layer, but the principle still applies. Mixing up the fund companies helps minimize your exposure should one of them do something stupid with your money.
There's also an argument to be had for splitting your money between brokerages. Even if you totally trust Vanguard as a company, it may not be wise to have only one login between an identity thief and your life savings. Having access to half your money at Fidelity or elsewhere would be very helpful while you sort things out.