In the United States, a paper railroad is a company in the railroad business that exists "on paper only": as a legal entity which does not own any track, locomotives, or rolling stock.
It allowed incorporation of a company and the seeking of capital to build a proposed railroad.
[1] Throughout the 19th and early 20th centuries, this specific connotation of the phrase "paper railroad" was consistent: a proposed, often speculative (and sometimes wildly speculative) venture in which a company stock exists, but no physical assets to run a railroad do.
In many cases, these railroads still existed as corporate entities long after plans to build them had been scrapped.
Reasons railroads are operated this way include those of internal corporate structure, tax ownership or locale, or for public relations purposes.