Tokens are unique to each mint. Every time you leave a mint over lightning you reveal amount + destination to the original mint (a potential point of de-anonymization/reduced anonymity). You're also narrowing down your anon set the larger value of tokens you hold (ecash tokens have "buckets" of 1,2,4,8,16, 32...etc, increasingly fewer denominations exist the larger the bucket)
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I read somewhere that you can just give the ecash token to a second mint and they can fetch the amount from the first mint.
In this way you could move funds between mints, and even spread funds around, without revealing your identity or amount held.
This is only with fedimint and not cashu (as of right now; they keep open the option of building federated mints but it is not currently planned).