In some openings, i.e. the Stonewall Attack / Maroczy Bind, the opponent gets an iron grip over a particular square (e5 / d5) by controlling it with two pawns, and they can stick a knight there. People are often very reluctant to trade this knight off, even if it's a very powerful piece that controls a lot of squares in the opponent's territory. In concrete terms, why would trading off for the knight (via capturing on e5 / d5) generally be bad for Black?
Then only strategically relevant reason, that could reasonably be attached to the concept you presented, is in my opionion, that the subsequent change of the pawn structure is unfavourable for the defender. Reasons would vary wildly with the concrete pawn structure. E.g.:
In the maroczy structure white may play Nd5 after preparation and if black captures the knight, white can capture with the e-pawn and make the e7 pawn backwards. Or white captures with the c-pawn, opening the c-file because he suddenly can invade over it.
In the stonewall structure black plays often Ne4 and if white would take, black would normally recapture with the f-pawn, having a half open f-file and driving a knight on f3 away. This may lead to an attack or if not that could force white to take prophylactic measures with f3, leading possibly to simplifications. Hence white often avoids the exchange altogether and prepares to kick the knight with f3 and a further e4.
If white has an isolani on d4, he often play Ne5 or even Nc5. If the knight is taken on c5, this leads to a passer. If the knight is taken on e5, white strenghten his pawns and again may become attacking chances.
Your questions makes absolutely sense to me, but if you want to study chess seriously you can only answer it in on a case by case basis. Stockfish is your friend.
Old FM Roese at your service (https://lichess.org/@/fiskaren)