WASHINGTON—The House Republican tax bill intensified a debate about whether the GOP is setting up a big giveaway to the rich, but there is a broader policy shift embedded in the proposal: What matters isn’t just how rich you are, but how you are rich.

The plan, which was unveiled this past week and will be further dissected when lawmakers return to Washington on Monday, aims to eliminate estate taxes by 2024. It cuts to 20% the tax rate on corporate income. It also cuts to 25% the rate on some income that owners of so-called...