Genuine question; If they're competitors, why is Vanguard Blackrock's biggest shareholder?[0] Is this the norm? Or is there a link between the two? (I don't know anything about the stock market)

[0] https://finance.yahoo.com/quote/BLK/holders?p=BLK

Vanguard is probably biggest shareholder of many companies, because the index funds it runs are huge.

The second largest shareholder of Blackrock, appears to be Blackrock, notice?

How do you think it is that Blackrock only owns 6.79% of Blackrock?

When you figure that out, you will have achieved enlightenment and be able to answer your original question.

What's the relative scale of the two?

Comparable. If the front page of Google is correct, Blackrock had 10T and Vanguard 8T assets under management at the start of the year

Check out Euroclear.com. They exclusively handle all back end clearing and settlement on the Paris, Brussels, and Amsterdam Bourses, as well as the London Stock Exchange. They may now handle some other stock exchanges, too. They also handle the majority of backend clearing and settlement across Europe.

As of the last time I checked (several years ago), they had over 27T Euros of assets under management.

Their competitor in the US is Depositary Trust Corporation (DTC), and last I checked they had over one quadrillion dollars of assets under management.