Today, we have a query of an issue not previously argued.
I have website "TheNameThatIHave.com". Today, I discovered that somebody else is publishing to "TheNameThatIHave.blogspot.com". That's not right - I should have the right to "TheNameThatIHave.blogspot.com"!
And the answer here is
No, if somebody else has "TheNameThatIHave.blogspot.com", it's their URL. You have no claim to "TheNameThatIHave.blogspot.com", simply by having "TheNameThatIHave.com".
There are multiple top level domains on the Internet, as well as various country based top level domains, and numerous web hosts similar to Blogger. Each domain / host has their own name directory, and assigns / allows assignment of names from its own universe. Assignment of any name in one top level domain / sub domain directory is irrelevant to eligibility to the same, or similar, name in any other directory.
There is no screening, to determine the validity of anybody to publish to "TheNameThatIHave.blogspot.com", for any value of "TheNameThatIHave". If the URL "TheNameThatIHave.blogspot.com" is available for publishing, every blogger has equal claim to take it. If you see it, and you take it, you have it - for eternity. The moment you take it, nobody else has any claim over it - for eternity. If somebody else took it, you have no claim over it - even if you own "TheNameThatIHave.com".
As the well used adages go
First come, first served.and
You snooze, you lose.