What does a person “with about two dollars to one’s name” mean?

Google shows over a hundred thousand references for the quoted phrase "two dollars to his name". Some of those references are bogus (e.g. two as part of thirty-two) but most of them that I looked at seem to entail the same meaning as in the subject quote: just about broke, having maybe a dollar bill or two, maybe some small change.

In short, "two dollars to his name" is an American colloquialism, a set phrase that indicates a person is financially busted.

As to the "nameless person with no value attached to his or her name / being" idea: No, as I understand the phrase it's entirely about financial status, and says nothing about the person's value or worth as a person. The phrase (as a whole) is not useful for other purposes than that, I think. Of course the two main parts of it (two dollar and to his name meaning, owned by him) are easily adapted. However, for terming something worthless, two cents is far more likely.

Here are a few examples from among the Google results mentioned above:

...he owned hundreds of acres of prime timber land but you couldn’t tell by looking at him that he had more than two dollars to his name. - Tim George, 2009
When he reached Pittsburg he had but two dollars to his name. - The Ariel, 1827
He wouldn't admit it to Jupiter, but he only had about two dollars to his name. - The Gun, O. C. Judd, 2000
He was a Russian immigrant who came to the United States in 1913 with two dollars to his name. - Barry Popik, 2005
With two dollars to his name, Ponzi emigrated from the U.S. to Canada... - ScoundrelsWiki, 2008
He had two dollars to his name, so one dollar went to pay for the marriage license and the other dollar went to pay the preacher. - Glenwood Resident Celebrates 99th Birthday, 2011

To one's name means in one's possession. The article is stating that Ray Kachel owns only two dollars. The use of two dollars in the quote is not idiomatic, it refers to a literal two dollars; to one's name can be used with other items (e.g. She has only a nickel to her name, or The only things he has to his name are the clothes on his back).

A partial answer: several years ago I recall reading about some places with laws that made it illegal to be poor and $2 was the figure quoted - if you don't have more than two dollars in your pocket you can be arrested. The reasoning given for the law was it gave the police the ability to take a drunk person off the streets, give them a meal/bath/bed for a night and the judge would typically release them the next morning. IIRC the article was about the demise of such laws.

In the US, two dollars is a small sum of money, typically slightly less than the cost of a loaf of bread or carton of milk, what might be pocket change for many people. There's nothing special about the exact quantify of money listed in the expression, I think it's equally common to say that someone doesn't have "so much as a dollar to their name". In either case, the implication is extreme poverty, lacking the financial resources to get by, to the point where they may not even have enough money for their next meal. This would be extreme contrast to say, a millionaire with ample financial resources.

And as you presumed, neither of the definitions you found through Google have any relation to the expression in the original article.