No, not really. It is specifically not a single country.
Most generally, this is the entire Arabian Peninsula, which currently includes the countries of:
- Saudi Arabia (most of it)
- United Arab Emirates
The root of the word is “Arab,” which has meant all sorts of things over the years, including “desert” and “nomad.” This fits with the ancient history of the general area being a desert inhabited by nomadic wanderers. Perhaps they came to be called “Arabs” and the area in which they wandered came to be called “Arabia.”
But different people in different times have used the word “Arabia” to refer to different subdivisions of the peninsula, and sometimes even north into mainland Asia.
The country of Saudi Arabia effectively “claims” the name by being the largest by far and having “Arabia” in its name, but their claim is nothing special. There was a man named Saud ibn Muhammad ibn Muqrin, who became the patriarchal head of the dynasty, which became known as The House of Saud.
Saudi Arabia literally means “the Saudi Arab kingdom,” or, effectively, “the part of Arabia controlled by the Saud family.” Theoretically and linguistically, if the royal family were to be overthrown, the country would need to be renamed. It’s only been in formal existence since 1932, and the area has been inhabited for millennia.
I actually found a Quora question called “Does all Arab ancestry originate from Saudi Arabia?” Some of the answers seemed to imply some annoyance from (presumably) non-Saudis:
[…] no first of all Saudi Arabia is a modern day country and not a good scale for what you’re talking about
[…] Various DNA tests have been run on members of the Saudi royal family and the result is that genetically, they are Indians from India.
[…] Yes all Arab originated from the Arabian peninsula I don’t like to use the name Saudi Arabia because it is a recent country that been only for a 100 year [sic]
[…] It is not possible to know who came to Saudi Arabia only by the name of the family (tribe)
Since Arabia is a peninsula, the east, south, and west borders are settled. But how far north does it go?
The northern portion of the peninsula merges with the Syrian Desert with no clear borderline, although the northern boundary of the peninsula is generally considered to be the northern borders of Saudi Arabia and Kuwait.