It's stated clearly in the footnote in the manual as well as the Act itself. For a US "person", a US Tax ID is required.
Footnote 48 in BSA Manual:
An identification number for a U.S. person is a taxpayer identification number (TIN) (or evidence of an application for one), and an identification number for a non-U.S. person is one or more of the following: a TIN; a passport number and country of issuance; an alien identification card number; or a number and country of issuance of any other unexpired government-issued document evidencing nationality or residence and bearing a photograph or similar safeguard. TIN is defined by section 6109 of the Internal Revenue Code of 1986 (26 USC 6109) and the IRS regulations implementing that section (e.g., Social Security number, individual taxpayer identification number, or employer identification number).
CIP Section of BSA:
(i) Customer information required. (A) In general. The CIP must contain procedures for opening an account that specify the identifying information that will be obtained from each customer. Except as permitted by paragraphs (b)(2)(i)(B) and (C) of this section, the bank must obtain, at a minimum, the following information from the customer prior to opening an account:
(2) Date of birth, for an individual;
(3) Address, which shall be:
(i) For an individual, a residential or business street address;
(ii) For an individual who does not have a residential or business street address, an Army Post Office (APO) or Fleet Post Office (FPO) box number, or the residential or business street address of next of kin or of another contact individual; or
(iii) For a person other than an individual (such as a corporation, partnership, or trust), a principal place of business, local office, or other physical location; and
(4) Identification number, which shall be:
(i) For a U.S. person, a taxpayer identification number; or
(ii) For a non-U.S. person, one or more of the following: a taxpayer identification number; passport number and country of issuance; alien identification card number; or number and country of issuance of any other government-issued document evidencing nationality or residence and bearing a photograph or similar safeguard.
Note to paragraph (b)(2)(i)(A)(4)(ii): When opening an account for a foreign business or enterprise that does not have an identification number, the bank must request alternative government- issued documentation certifying the existence of the business or enterprise.
(B) Exception for persons applying for a taxpayer identification number. Instead of obtaining a taxpayer identification number from a customer prior to opening the account, the CIP may include procedures for opening an account for a customer that has applied for, but has not received, a taxpayer identification number. In this case, the CIP must include procedures to confirm that the application was filed before the customer opens the account and to obtain the taxpayer identification number within a reasonable period of time after the account is opened.