I'll be honest, I had not heard much about throat singing in Viking culture before seeing this question. It is very hard to say much about historical folk music because of the ephemeral nature of the medium, and even harder to say much about vocal music, because there aren't even artifacts of instruments to study.

The main reference to throat singing in Old Norse culture we have comes from a traveler from Al-Andalus (Muslim Spain), Ibrahim ibn Yaqub al-Tartushi. He was, shall we say, not a fan...

I have not heard an uglier singing than the people of Schleswig. It is a humming coming from their throats that's worse than dogs barking.

It's good to define what we're talking about. First off, we're talking about people in what's now Southern Denmark/Northern Germany. I am in no way, shape, or form a scholar of Old Norse culture, but it does seem like the people living there could be classed under the "Viking" label. How similar they were to the Vikings of what's now Norway, Sweden, etc., I have no idea.

Second, it isn't quite clear what exactly he's describing, and whether it would fit into the modern concept of "throat singing," which usually describes a method of singing where the singer manipulates overtones with their throat, producing a prominent whistling-like sound above the note that they are singing. However, it does sound very promising. "A humming coming from their throats" is a pretty good description of what you'd hear from Tuvan or Mongolian singers today. I'll leave the music criticism to Mr. al-Tartushi, but the gruff sound of the fundamental note that is usually used in throat singing could be reminiscent of dogs barking, which would make his second statement also relevant beyond his own personal taste in singing.

All that out of the way, the broad answer to your question is that throat singing today is not confined to the Tuvans, Mongolians, and other peoples of Central and East Asia. We see it in South Africa, Sardinia, and the area where Iran, Pakistan, and Afghanistan all meet. Similar techniques, often involving a duet between women, are found in Japan and among Inuit cultures. There is even some throat singing among the Sami people of Northern Scandinavia, which shows that it has persisted to this day in that region (unfortunately, I was unable to find any clips of Sami throat singing).

So, tldr; throat singing is actually fairly common, even if certain styles have become much more widely known than others. Because of the difficulty of determining what the singing of the past sounded like, I can't say for certain what the Celts and their contemporaries did or didn't do, although you might be interested in an answer I wrote here a while back about what we call "Celtic music" today.

And just because it's an epic video in so many ways, I'll post another clip of one of my favorite throat singers, Batzorig Vaanchig.