A pen name is absolutely about branding. Not anonymity.
Not necessarily true. Sometimes it's about anonymity as well -- for various reasons.
Of course, it's "mostly" marketing and branding, sure, but to say "absolutely" is false, there's romance writers who totally want to stay anonymous due to the nature of the things they write and there's also people who, for various reasons, don't want to be found out, say ex-spouses or the government or whatever.
It's not exactly hard to set up an LLC or even a foreign corporation and keep your real identity hidden from Amazon, examples: Bella Forrest and Russell Blake.
In some cases the pen name is trademarked for reasons to do with tax liability or other personal reasons to stay anonymous -- it's also done for branding and marketing.
For example, to run Sponsored Brands ads (and Amazon Stores) you must have a registered trademark (which doesn't have to be the pen name, it can be the name of your publishing company if you want, but using the pen name kills two birds).
If any wants to see what pen names are trademarked, it's a simple search --- and another trick to see who is offshoring is to look at the newsletters they send out and see if the address is Cypress or some other foreign entity used for tax purposes.