To expand just a bit further sdsearch said, one of the more common instances of the two-browser trick being used was the Citi AA personal cards. Citi issued an AA Visa and an AA Amex, both essentially the same card, but just on a different network.

Having one would disqualify one from getting a sign up bonus on the other. So using the "two-browser trick" would make Citi think you are a first time cardholder on EACH card, getting you the sign up bonus on BOTH cards. This I thought was the main benefit of the two-browser trick, to double your sign up bonus.

Yes, usually the credit bureaus think it's an identical application so the pulls get merged, but I always considered that a secondary benefit to the extra miles (or rather, a lack of a con because you wouldn't get an extra pull for doing this for the extra miles)