Is there a name for the abbreviated (usually three syllable) title?

Question

Surely the act of abbreviating an anime title to three syllables has a name...? And what is that name? (some of these will show my age)

╔═════════════════════════════════════════════╦═══════════╗
║                  Full Name                  ║ Shortened ║
╠═════════════════════════════════════════════╬═══════════╣
║ Kodomo no Omocha                            ║ Kodocha   ║
║ Jungle wa Itsumo Hare nochi Guu             ║ Hareguu   ║
║ Ore no Imouto ga Konnani Kawaii Wake ga Nai ║ Oreimo    ║
║ Fruits Basket                               ║ Furuba    ║
║ Japanese Animation                          ║ Anime     ║
╚═════════════════════════════════════════════╩═══════════╝

Answer 1

For what it's worth, in Japanese, you would call this particular sort of abbreviation a 略称 ryakushou or 略語 ryakugo, both of which basically just mean "abbreviation". There isn't an anime-specific term for this sort of thing or anything.

I should mention that abbreviated names of anime are more commonly four syllables than three-syllables, at least in recent years. "Hareguu" is Ha-re-gu-u (actually, these are morae rather than syllables, which is why this may not be obvious if you're unfamiliar with Japanese); "Oreimo" is O-re-i-mo; and Fruits Basket is frequently "Furubasu" Fu-ru-ba-su in Japanese. There is a nice table of these on Nicopedia (JP).

As a side point of interest, this blog (JP) collates abbreviated names for anime of recent seasons - for example, 2015 winter, spring, summer, and fall. (Obviously, machine translation is going to fail spectacularly for this sort of thing, so this is only relevant if you can read some Japanese.)

Answer 2

The name for the abbreviated word resulting from the combination is a Portmanteau

A portmanteau is linguistic blend of words in which parts of multiple words, or their phones (sounds), and their meaning are combined into a new word.

A very common type of portmanteau that is used within the Japanese language is forming one word from the beginnings of 2 other words.

A Sino-Japanese example is the name 東大 (Tōdai) for the University of Tokyo, in full 東京大学 (Tōkyō daigaku). With borrowings, typical results are words such as パソコン (pasokon), meaning personal computer (PC), which despite being formed of English elements does not exist in English; it is a uniquely Japanese contraction of the English personal computer (パーソナル・コンピュータ pāsonaru konpyūta?). Another example, Pokémon (ポケモン?), is a contracted form of the English words pocket (ポケット poketto?) and monsters (モンスター monsutā?). A famous example of a blend with mixed sources is karaoke (カラオケ karaoke?), blending the Japanese word for empty (空 kara?) and the English word orchestra (オーケストラ ōkesutora?). - source

As for the act of abbreviating, it is sometimes revered to as linguistic word blending or word contracting and more commonly known as abbreviating. The amount of syllables has no effect on the terminology used to describe this act.