The exact term for such a group of three digits is a period. Thus you have the ones period, the thousands period, the millions period, etc. This is one of those wonderful terms we all learned (and then forgot) in primary school.
The term is in current use but hard to find in general purpose dictionaries. For modern examples from online arithmetic textbooks and study aids, see:
It is not of modern origin; the word period has carried this meaning for hundreds of years. Here is the relevant sense of period from NED (the predecessor of the OED, now public domain).
Period (pīᵊ·riŏd), sb. Forms: 5 peryod, paryode, 6 peryode, periode, 6– period: see also PARODY sb.² [a.F. période (14th c. in Hatz.-Darm.) = Sp. periodo, It. periodo, ad.L. period-us, a.Gr. περίοδος going round, way round, circuit, revolution, cycle of years, periodic recurrence, course, recurring fit of disease, orbit of a heavenly body, rounded sentence, f. περί around + ὁδός way; in ancient L. used only of the period or cycle of the four Grecian games, and of a complete sentence; in med.L. in other of the Gr. senses.]
III. In Grammar, Rhetoric, Music, etc.
13. Arith. A set of figures in a large number marked off by commas placed between or dots placed over, as in numeration, circulating decimals, and the extraction of the square or cube root.
1674 JEAKE Arith. (1696) 15 A Period is a comprehension of Degrees .. as 123 .. 12345, &c.
a 1677 Cocker’s Arith. i. § 10. 6 A Period .. when a Number consists of more than three figures or places, whose proper order is to prick or distinguish every third Place .. so .. 63.452.
1690 LEYBOURN Curs. Math. 4 Numbers .. of Three Figures, or Places .. may properly be called a Period.
1704 J. HARRIS Lex. Techn. I. s. v., A Period in Numbers, is a Distinction made by a Point, or Comma after every sixth Place or Figure; and is used in Numeration, for the readier distinguishing and naming the several Figures or Places.
1859 BARN. SMITH Arith. & Algebra (ed. 6) 76 The part [of a circulating decimal] which is repeated is called the Period.
—A New English Dictionary On Historical Principles (1888)¹