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What are the general characteristics of a pawn-like piece? I'd nominate
these characteristics:

1. Most numerous piece type in the game.
2. Weakest piece type in the game.
3. Short range.
4. Non-retreating.
5. Promotes to something decisive (can force mate).

For illustration, consider how several variants stack up:

FIDE Chess: pawn satisfies each criterion perfectly.

Shatranj: Perfect for 1-4, deficient in criterion 5, as K + Ferz that a
pawn promotes to can't force mate (less of a problem with the Shatranj
ruleset as stalemate and bare King are wins).

my own Pocket Mutation Chess: 1-3 is perfect, 4 is not so much so, as a
pawn can retreat via a pocket move, 5 partially not a fit, while the pawn
has a promotion path to a decisive piece, it can only only promote directly
to a Knight or Bishop, which can't force mate.

Betza's For The Birds Chess: 1, 2, 4 and 5 OK but the pawn-like piece has
a long range move.

my own Wizards' War: nothing remotely resembling a pawn in this game (by
design--one of my design objectives was a playable, pawn-less, strong piece
game).

I submit that all the games are playable Chess Variants (broadly defined)
but the better a variant conforms to these criteria, the more
'Chess-like' it is.

Try analyzing some other variants with these criteria and let me know what
you think of this hypothesis, offering alternative/additional criteria if
you wish.