So far as Anglo is concerned, it originates from post-classical Latin, the earliest examples being from the late-sixteenth century. Perhaps the earliest such combination was Anglo-Saxon, meaning an English person of Saxon origin.
The OED has a substantial reference on Anglo-Saxon with a lot on the etymology. It follows the classical Latin and Greek form of adjectival connection, such as sacrosanctus - "sacredly sanctified".
All of these forms, such as Franco, Germano, Indo etc are undoubtedly borrowings from Latin.
Of course the term Anglo-American has more than one meaning. One can speak, for example of an Anglo-American corporation, or let's say Anglo-American ideas etc meaning a combination of the two. There is however another way in which Anglo-American is used - meaning an American of English origin. This is a much later sense, and appears in the nineteenth century.
Anglo-Indian can mean either a person of mixed English and Indian ethnicity, or it could mean an Indian of English origin i.e. typically during the Raj, a white person settled in India.
Etymology: < post-classical Latin Anglo-Saxones the English people,
sometimes (in British sources) apparently spec. the West Saxons
together with the Mercians (860, a1010 in continental sources; from
10th cent. in British sources) < Anglo- , combining form of classical
Latin Anglus (see Angle n.3) + post-classical Latin Saxones , plural
of Saxon- , Saxo Saxon n.; compare the earlier collocation Angli
Saxones , Saxones Angli , lit. ‘the English Saxons’, the English
people, in contradistinction to the continental Saxons (8th cent. in a
continental source). With use with reference to Old English compare
earlier Saxon n. 2. In post-classical Latin Anglo-Saxones , Anglo- ,
combining form, is used adverbially, as in similar Latin and Greek
compounds, e.g. classical Latin sacrōsanctus sacredly sanctioned (see
sacrosanct adj.), Hellenistic Greek Ἰνδοσκυθία Indian Scythia, Scythia
of the Indus (compare Indo-Scythian adj. at Indo- comb. form1 2),
Hellenistic Greek Συροϕοῖνιξ , classical Latin Syrophoenix ,
Phoenician of Syria (see Syrophœnician n.). Compare also classical
Latin Gallograecī (see Gallo-greeks n. at Gallo- comb. form1 1b).
Post-classical Latin Anglo-Saxones , which had become rare after the
Norman Conquest, was apparently revived in historical use by Camden
(writing in Latin) in the late 16th cent.: 1586 W. Camden Brit. 43
(running title) Anglo-Saxones. 1607 W. Camden Brit. (rev. ed.) 94
Nunc..Anglo-Saxones ad differentiam eorum in Germania, vocatos. 1607
W. Camden Brit. (rev. ed.) 128 Maiores nostri Anglo-Saxones
Wittena-ge-mott, .i. Prudentum Conuentus..vocârunt. Compare earlier
English-Saxon adj. Compare also earlier use of Angle-Saxon denoting
Old English ( < Angle n.3 + Saxon n., after post-classical Latin
Anglo-Saxones): 1589 G. Puttenham Arte Eng. Poesie (new ed.) iii.
iv. 120 I meane the speach..so is ours at this day the Norman
English. Before the Conquest of the Normans it was the Anglesaxon, and
before that the British.