I recently had my 10 year old 50 gal Rheem hot water tank replaced with a 50 gal Bradford White - both natural gas. The problem with the new tank is when one of us (family of 5) is taking a shower and another hot water valve is opened, the hot water pressure in the shower drops, which results in a cold shower. This never happened with the old water tank. Any suggestions?

The only time this issue occurs is when water lines on the bathroom side are undersized when roughed in(installed). What is happening is when your mixing hot and cold to get a warm shower and the hot is turned on the faucet immediately, as a result it is drawing the hot from your tub and you get a surge of cold. Fixtures are sized on fixture units, length, location of water source.
Or your hot water pressure on your shower is low as a result the faucet is drawing hot from shower and get cold surge.

If this were the case, why didn't it happen with the old hot water tank? Also I used the example of the shower, the drop in pressure could be any faucet in the house when another hot faucet is turned on.

No, it would not happen if the main trunk is size correctly. A lot of times is when branching off to fixtures where it occurs because for instance water closet is 2.5 fixture units and a shower is 2 units which if size correctly it should be 3/4" supply and not 1/2". It just an example.
Try to flush your hot lines where u are having the issue.

Increase water pressure... and remove & clean your shower head. See if that helps