The 3 most recent kids I know without a connection to either school who gained admission were admitted to both.
Almost everyone I know who was admitted to Harvard and not Stanford was a legacy (including kids who were legacies at both. This is partly because legacy status counts for less at Stanford. )
But in some ways, that is more of the Stanford culture - a little more “California modern” than “traditional”. Super accomplished kids, very qualified, but hooked in that way.
And I know tons of kids who applied to (and were admitted to) only one of the two. Many were also legacies. They are pretty different schools in terms of vibe.
In terms of history, Stanford was really much more of a regional school through the 70s. If you were a high achieving West coast kid, you applied to Stanford. It had great programs and was well regarded among its peers, but it was not a household name like Harvard.
I don’t think either school has a “type” they admit, but the institutional priorities can affect applicants differently.