Bitcoin fees are charged per unit of data, not per unit of value, so a large transfer can cost less than a small one. The unit used is virtual bytes, which requires explanation.
Block space is the scarce resource
A block can hold a limited amount of data, and transactions compete for inclusion by offering fees relative to how much room they occupy.
The amount transferred is irrelevant to that competition, since moving a large sum consumes no more space than moving a small one.
What determines size is structure: how many inputs are being spent, how many outputs are created and what conditions must be satisfied.
Inputs dominate the size
Each input references a previous output and carries the data proving the spender may use it, which is the largest part of most transactions.
A wallet spending many small received amounts must include an input for each, producing a large transaction and a correspondingly large fee.
This is why fee estimates depend on which coins a wallet selects, and why the same payment amount can cost very different fees from different wallets.
Segregated witness separated two kinds of data
The upgrade moved signature data into a separate section of the transaction, which changed how that data counts toward the block limit.
Witness data is discounted relative to other data, reflecting that it is not needed for the long-term record of which outputs remain unspent.
Weight units express this, with the discount applied so that a transaction's cost against the limit accounts for the different burdens the data imposes.
Virtual bytes are weight divided down
Virtual bytes convert weight into a figure comparable to the pre-upgrade size measure, so fee rates could continue being quoted in familiar terms.
A transaction's virtual size is therefore smaller than its raw byte count whenever it contains witness data, which is most transactions today.
Fee rates quoted per virtual byte are what wallets and estimators use, so comparing rates requires knowing which unit a figure refers to.
Address type changes the arithmetic
Different address formats produce inputs and outputs of different sizes, so the choice affects fees on every transaction the wallet ever makes.
Newer formats generally produce smaller virtual sizes, which is why wallets migrated to them beyond any feature considerations.
The saving compounds for anyone transacting frequently or spending many inputs, and it is the practical reason address format is discussed as an economic decision rather than a cosmetic one.