The categories used by the federal government are specified by the Office of Management and Budget (OMB) in its Statistical Policy Directive 15, "Standards for Maintaining, Collecting,
and Presenting Federal Data on Race
and Ethnicity", 62 FR 58782. The directive was last updated in 1997. The standard itself is at the end of that document (bottom of page 58788), and several pages of background and rationale are also included, as well as information about the process by which the standard was developed.
The directive specifies in Section 3 of the standard that it is to be used for essentially all federal and federally sponsored purposes. It might also be used voluntarily by state and local governments or other organizations, to make it easier to correlate with federal data.
The OMB is an executive branch agency, part of the Executive Office of the President (i.e. "the White House"), and this directive seems to have followed the usual federal rulemaking process used when executive branch departments create regulations (overview). In principle this is supposed to happen pursuant to laws passed by Congress, but I wasn't immediately able to tell which laws those were. It might be that some statutes include a general requirement to collect statistics on race and ethnicity in certain situations, and the government infers from this that it has the authority to create standards on how to do so.
Obviously the standards are controversial and many people have questions or concerns about the categories that were chosen. The OMB began a limited review of the standard in September 2016, see 81 FR 67398, and there is a docket with about 1200 public comments they received. I'm not sure whether the review was ever completed (the subsequent change in presidential administration may have had something to do with that) but in any case it does not appear that any revisions were made.
The distinction between ethnicity and race seems to predate the 1997 directive, perhaps coming from the previous version of 1977. The 2016 review notice cites this as "42 FR 1926 May 12, 1977", but I haven't been able to find its text yet (and the date doesn't seem to be consistent with the page number).
I don't think any version of this document makes any statements such as "Dominicans shall identify as X". It is always left to each individual respondent to decide which classification they believe best fits their self-identification.