Sometimes, with caveats (I researched this a while back). tl;dr: probably yes, if you get puzzles manufactured by the same company around the same time.
There are “puzzle companies” that are actually just design studios who farm out the manufacturing process to a puzzle printer. Depending on the printer, they’re very likely to reuse puzzle dies even between design companies; so you might even be able to remix two puzzles by different companies, if they use the same print company.
Certain high volume printers rebuild their puzzle dies from scratch every so often (so the design would be entirely different depending on which print run you’re working with). Ravensburger in particular had a thing on their website (it might still be there) about how they can’t get you individual replacement pieces because they’ve almost certainly already rebuilt the die since your copy was printed. And most companies end up just replacing the entire puzzle if they leave out a piece.
Some puzzle companies make puzzles where the design and the cut are related somehow (Magic Puzzle Company built a whole line on those). Those are unlikely to be reused between designs, though you do end up paying extra for them.
Even if you get a pair of puzzles that were cut using the same die, they might not line up the way this image shows. If the die was removed and flipped around between one puzzle’s cutting and another, the cut would be “upside down” on the other and they would fit together in a very different way.
Sometimes, with caveats (I researched this a while back). tl;dr: probably yes, if you get puzzles manufactured by the same company around the same time.
There are “puzzle companies” that are actually just design studios who farm out the manufacturing process to a puzzle printer. Depending on the printer, they’re very likely to reuse puzzle dies even between design companies; so you might even be able to remix two puzzles by different companies, if they use the same print company.
Certain high volume printers rebuild their puzzle dies from scratch every so often (so the design would be entirely different depending on which print run you’re working with). Ravensburger in particular had a thing on their website (it might still be there) about how they can’t get you individual replacement pieces because they’ve almost certainly already rebuilt the die since your copy was printed. And most companies end up just replacing the entire puzzle if they leave out a piece.
Some puzzle companies make puzzles where the design and the cut are related somehow (Magic Puzzle Company built a whole line on those). Those are unlikely to be reused between designs, though you do end up paying extra for them.
Even if you get a pair of puzzles that were cut using the same die, they might not line up the way this image shows. If the die was removed and flipped around between one puzzle’s cutting and another, the cut would be “upside down” on the other and they would fit together in a very different way.
Pretty cool stuff