Metal strings are primarily made with steel which has a high Young’s Modulus, meaning it is very stiff and resistant to stretching. When you press a steel string down to a fret, you are slightly increasing its length. Because steel does not stretch easily, this tiny increase in length causes a disproportionately large jump in tension, which pulls the note sharp.

Nylon and Gut Strings have a much lower Young’s Modulus; they are highly elastic and "stretchy". When you press a nylon string to a fret, the material absorbs the added length through stretching rather than increasing in tension. Consequently, the pitch remains much closer to the intended note without needing intonation adjustments.

Physics dictates that a "perfect" vibrating string should be infinitely flexible. Metal strings have internal stiffness that affects how they vibrate.

Stiffness causes a string to act like a rigid rod near its anchor points (the nut and bridge). This makes the string vibrate as if it were shorter than it actually is, a phenomenon that creates inharmonicity or overtones that are sharper than they should be.

Because metal is much denser than nylon, steel strings are thinner for the same pitch. However, it’s the core wire of a wound steel string is what determines its tension and intonation behavior not the “outer” dimension of a wound string.
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String Material and Construction: Every string brand is different. Even at the same tension, different materials (e.g., nickel vs. steel) or core types (hex vs. round) have different levels of flexibility and "inharmonicity," which changes how they vibrate and dictate the necessary intonation adjustments.

It’s important to remember that new intonation adjustments are needed when changing string tension (buy changing tuning or string gauge) because any change in the force pulling on the instrument alters its physical geometry and the way the string reacts when fretted.