I could be wrong, but it seems like the major determinant of whether a piece is a major piece or a minor piece is whether a king and one of that piece can force checkmate on a lone king. Would this be possible with a king and one of these hypothetical pieces? (Or is that not actually the determinant of what is a major piece and what is a minor piece?)
This piece is widely used in different fairy chess variants. It appears under several names, including Commoner, Guard and Man / Mann.
While the table in the Wikipedia article also linked to in @Allure's answer states it's worth about four points, the article itself limits that value to the endgame:
In the endgame, where there is usually little danger of checkmate, the fighting value of the king is about four points (Lasker 1934:73). In the endgame, a king is more powerful than a minor piece but less powerful than a rook. Julian Hodgson also puts its value at four points (Aagaard 2004:12).
On the other hand, the Wikipedia article about the fairy chess piece says
A man is approximately equal in strength and value to a knight, generally.
That would make it a minor piece. It might depend on the board size (it's less effective on large boards, and worth much more on small boards) or the other pieces available to the players.
it seems like the major determinant of whether a piece is a major piece or a minor piece is whether a king and one of that piece can force checkmate on a lone king
That's close to the definition I've heard: a major piece is one which can cut off the enemy king from one side of the board. Rooks and queens can do this, but this non-royal king cannot.
On the other hand, there is a basic situation where this piece is even stronger than a rook. King + rook lose against king + queen, but king + commoner is a rather simple draw.
I haven't heard of that definition being used for a major piece, but it's a neat observation. Such a piece (call it X) should be able to force a checkmate in a K+X vs K situation.
Say White's pieces are on d3 and e3, while Black's is on d5:
1) White plays Xd4, and Black's king must retreat. Say ...Kd6 is played.
2) White plays Ke4. Now Xd5 is coming, forcing Black's king further back. ...Ke6 doesn't help him since it runs into Xe5. So Black could try ...Kc6, as White's king doesn't hold it back there.
3) Now instead of advancing the 'X' piece to d5 (since it would allow ...Kb5, although this would also work nonetheless), White could play Xc4, placing Black in zugzwang. In the case of ...Kd6, White plays Xd5. And in the case of ...Kb6, White plays Kd5, forcing Black further towards one side.
This general process can be repeated.
I don't know of a strict definition of what makes a major piece 'major'. Of the queen, rook, bishop, and knight, a major piece is just one that is in the top half strength-wise. So a king-like piece shouldn't be considered a major piece, since typically it only reaches the rough strength of a bishop/knight in sufficiently simplified endgames.
EDIT - one thing to note is that, because capturing this hypothetical piece wouldn't result in the game ending, it's usefulness would be higher than a normal king. It could be safely used in the middlegame like any other piece. So it's possible it could be stronger than a minor piece, but still below a rook.
The king is worth about 4 pawns if its loss didn't lose the game, which means it's neither a minor or a major piece, but something in between.
(Major pieces like the rook is worth 5+ pawns; minor pieces are worth 3.)