You don't have to buy D2 for cross-save. In September, New Light launches for all players, which gives you free access to the base game and first 2 expansions on every platform. If you currently own Forsaken, the Forsaken annual pass, and/or are purchasing Shadowkeep, you will need to purchase those same licenses on the other platform IF you want to play the same content on both platforms. All of your gear will still carry over from cross-saves and you'll still be able to play without Forsaken or Shadowkeep licenses on your new platform. You'll just be locked out of story missions and a few activities. For example, you would be able to use Dreaming City gear, but you would not have access to the Dreaming City on the new platform you cross-saved to unless you buy Forsaken.

You only have to do that if you are playing on another platform. Platforms take up to a 30% cut from the sale of the game/dlc/expansion to use their platform, if you are playstation and want to play on xbox, for you to use that same license on Xbox, Sony would need to pay Microsoft.