The fact that the DoE acknowledges TRACS doesn’t mean other schools do, and in fact most don’t. TRACS is a religious accreditation organization and they work quite differently from academic accreditation.
TRACS accredits seminaries for certain religions. Seminaries by their definition are excluded from religious discrimination policies for obvious reasons (i.e. if FSU or Vanderbilt had a “no Jews, Muslims, or Catholics” policy they’d lose SACS accreditation in a heartbeat, but at a rabbinical college excluding Catholics and Muslims is a bit more valid as being Jewish is kind of important for a rabbi). The DoE, quite rightly, recognizes the ministry/missionary work/religion professors/etc. as a valid career and those studying it are eligible for Federal student loans, military benefits, etc., which is why Yes, the government does recognize TRACS.
People who attend a TRACS (or similarly accredited school) can apply to a SACS school (or whatever the accrediting authority is for most public and private colleges is in your region- the S in SACS is for Southern but there’s also Western, North Central, etc., and they all use pretty much the same guidelines and tiering) but said SACS school does not have to recognize their credits. If you’re applying for work in the humanities they may accept some credits (though they are under no obligation to)- English comp or some other very basic course perhaps, almost certainly no higher level courses. Apply to pharmacy or medical school with a biology or chemistry degree from BJU and you’ll be laughed at; you’re going to have to start at square one at a regionally accredited institution.
SACS and its related regional accrediting agencies are far more nationally and internationally prestigious and valuable for accreditation than religiously affiliated accrediation agencies because it means the school MUST meet certain criteria by way of core curriculum, library size, teacher:student ratio, standardized test scores of incoming and graduating students (big one), spending per student, etc… Mercer University in Macon GA {campuses in Atlanta and elsewhere as well} is a Baptist school, but they are also accredited by SACS (in addition to whatever Christian accreditation they use). By definition they teach evolution in their science classes, they are far more inclusive than most Southern Baptist churches (they have a GLBT student organization, for instance) and most importantly their credits transfer to wwwwwaaaaaaaaaaaaaaayyyyyyyy more institutions than would BJU’s or Jimmy Swaggart Bible College because their accreditation means they meet or exceed the national minimum academic standards for a university. (Mercer also has medical and legal schools that are accredited by the same agencies that would accredit the University of Georgia or UAB).
Short answer: there’s accreditation and there’s accreditation and they’re not equal.