Answer first: the practical privacy question is what a payment record reveals and to whom, and the answer differs by method more than most people assume.
Using a dedicated email address per merchant makes any subsequent breach traceable to its source and costs nothing to set up.
Keep your own record of every transaction regardless of method: date, amount, reference, and what was ordered. This is the documentation you will need if anything requires resolving.
Merchant payment-data breaches are a well-documented and recurring exposure across e-commerce generally.
A dedicated email address per merchant is free and genuinely useful.