There is "Full name" field on the user profile page:
Such name can be set different on each SE site. Is there a reason to allow that, since this name is not shown publicly?
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Ultimately, it's because there's no reason to forbid it either. Sure, it technically should be the same across the network. But keeping that field always in sync network-wide when it's barely used for anything is just not worth the resources required to do that. Nobody really cares what users type into the field, or if it's different across sites. So it's just a standard field that's... there.
The real name is used when the user is contacted by SE: automated emails or staff, e.g. I was contacted by SE swag provider who used my real name.
Since those are being sent via email, some people might find it easier to filter the emails when using a unique name for each site.
Also, nobody is forced to actually give their real name (SE is not Facebook ;)) so I'm pretty sure many users put their display name in there as well, which can be different on each site.