The relevant bit of federal law is Section 3394 of the Revised Statutes of the United States.
Effective July 24, 1897, as part of the so-called Dingley Tariff Bill, federal law prohibited any inserts of any kind to be packaged with any type of tobacco product. The statutory language was as follows:
"None of the packages of smoking tobacco and fine-cut chewing tobacco and cigarettes prescribed by law shall be permitted to have packed in, or attached to, or connected with them, any article or thing whatsoever, other than the manufacturer’s wrappers and labels, the internal-revenue stamp, and the tobacco and cigarettes, respectively, put up therein, on which the tax is required to be paid under the internal-revenue laws; nor shall there be affixed to, or branded, stamped, marked, written, or printed upon, said packages, or their contents, any promise or offer of, or any order or certificate for, any gift, prize, premium, payment, or reward."
Below is some good context and background on the law. This comes from testimony to Congress by John Yerkes, former Commissioner of Internal Revenue, in 1908:
The Dingley provision was challenged in the courts, all the way to the Supreme Court (Felsenheld v. U.S., 186 U.S. 126 (1902)), and upheld.
But shortly after the Felsenheld case, the law was repealed, effective July 1, 1902, and the language of Section 3394 was amended to read:
"No packages of manufactured tobacco, snuff, cigars, or cigarettes, prescribed by law, shall be permitted to have packed in, or attached to, or connected with, them, nor affixed to, branded, stamped, marked, written, or printed upon them, any paper, certificate, or instrument purporting to be or represent a ticket, chance, share, or interest in, or dependent upon, the event of a lottery, nor any indecent or immoral picture, representation, print, or words; and any violation of the provisions of this paragraph shall subject the offender to the penalties and punishments provided by section thirty-four hundred and fifty-six of the Revised Statutes."
The actual requirement to include the registered factory number and internal revenue district information on inserts is found in Internal Revenue Regulations. The earliest I've been able to pin down that requirement is from Regulations No. 8, revised August 1, 1907:
"In view of the report of the committee submitting the above amendment (act approved July 1, 1902), it is decided that a manufacturer may place within his statutory package containing tobacco, snuff, cigars, or cigarettes small advertising cards, coupons, certificates, paper bands, circulars, trade-mark tin tags, and trade-mark strips, and which do not materially increase the size of the package, and which are intended as an advertisement of his business, and concern the manufacture and sale of his products and no other business; and such advertising matter will not be prohibited, although intended to be returned to the manufacturer, or to some person designated by him on the coupon, and exchanged for other articles, provided the distribution of the prize articles does not depend upon the event of a lottery.
It is required that the manufacturer’s registered factory number, district, and State shall appear upon each card, coupon, or certificate placed within the statutory package."