feat: Generic unit on amount

[thesimplekid]

Description

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The Cashu protocol supports multiple currency units (Sat, Msat, Usd, Eur, etc.), but the previous Amount type was a simple u64 wrapper with no awareness of units. This created a subtle but dangerous class of bugs: nothing prevented code from accidentally adding 1000 satoshis to 500 millisatoshis, producing a nonsensical result of 1500 in an undefined unit. These bugs are especially insidious because the code compiles and runs without error—the incorrect math only manifests as wrong balances or failed payments in production.

This change makes unit mismatches impossible by encoding the currency unit into the type system. Amount now carries its unit at the type level, and arithmetic operations verify unit compatibility before proceeding. The compiler catches many mistakes statically, and runtime checks catch the rest with a clear UnitMismatch error rather than silent corruption.

The refactor maintains backwards compatibility by using Amount<()> (untyped) at serialization boundaries where the wire protocol expects a plain integer, while internal mint logic now uses Amount to get the safety guarantees. The with_unit() method provides a clear conversion point at the boundary between protocol parsing and application logic, making it explicit where unit context is being added.

This is particularly important for the mint's quote handling, where amounts flow between payment backends (which may use different units internally) and the core mint logic. By making MintQuote.amount_paid, MeltQuote.fee_reserve, and payment response types carry their units, we ensure that fee calculations, balance checks, and unit conversions are always performed correctly.

closes: #1463

• test with lnbits
• wait on feat: custom axum router #1251

Notes to the reviewers

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Suggested CHANGELOG Updates

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Checklist

• I followed the code style guidelines
• I ran just final-check before committing