The Bowery does take the definite article in common usage, even if not signed that way. It's a rare construction in the US (definite article + proper name), where we almost exclusively use proper name + designator (street/road/avenue etc.). It tends to give things an antique or old-world flair, though I'd wager that most current instances are newly fabricated just for that reason. The Bowery is one authentic example, at least.
In this vein, there is a street not far from where I live called "The Parkway." What I find amusing about it is that my Acura's sat-nav, which displays the name of a street without the descriptor (e.g., Duke Street shows up on the sat-nav map as "Duke"), displays that street as "The" unless and until you click the joystick somewhere along it to display an address or get directions. Google Maps displays it as "The Pkwy," which is similar in that it implies that the street is "The." So if you were just looking at the map on my nav screen, you'd think you had to turn left from "Kings" onto "The."
I wonder if the band "The The" ever visited that street.
Regarding "Bowery," the word is derived from a Dutch word meaning "farm." A few years ago for Christmas my mother gave me a copy of Edward Rutherfurd's novel New York
, which is historical fiction set in New York City beginning in the 1600s when it was still New Amsterdam and continuing up through September 11, 2011; as with his other novels, it follows a couple of fictional families. Anyway, one of the things that was interesting was seeing how many old Dutch names have become ubiquitous in ways we don't realize. The term "bouwerij" (sometimes anglicized to "bouwerie") was apparently a fairly common term back then used to refer to farmers' property in the area, especially as you went further north towards upper Manhattan or the land further north belonging to Jonas Bronck. Supposedly the street now known as either "Bowery" or "the Bowery" got its name in the same way that many other roads in the US got their names—it connected the small town of New Amsterdam to the farms ("boweries") located further out. (Same principle "kphoger" mentions in his comment further up the thread.)
I have absolutely no idea why the name has no descriptor, though. My gut suggests to me that the word "Bowery" is unique enough that people didn't feel the need to add anything to distinguish it and so the use of the descriptor just dropped away over time. "Broadway" at least works as an elision of "Broad Way" such that no descriptor would be needed.