Why was Shonen Jump called Shonen Jack in Bakuman Anime?
Because the anime was broadcasted on NHK.
NHK, being a public TV channel in Japan, doesn't allow commercial advertising.

Much like the BBC, NHK is funded by a &quot;receiving fee&quot; by every Japanese household, with no commercial advertising and the maintenance of a position of strict political impartiality.
(Emphasis mine)

Their official domestic program standard (Japanese) (or, &quot;standard policy&quot;) states this:

Item 12 - Advertising

It's absolutely not allowed to broadcast in purpose of commercial advertising or self-advertisement marketing.
In the middle of the broadcast, when it's planned to include corporate name, personal name, occupation, firm name, or trade (brand) name, regardless if it's an essential component or unavoidable performance, they will be judged impartially and then given the decision on how to handle them.


The staffs, trying to avoid this case as best as they can, replaced some of the names with others:

Shueisha (publisher of Shonen Jump) → Yueisha
Jump (Shueisha's manga magazine brand) → Jack
SHOSETSU SUBARU (Shueisha's monthly light novel magazine) → SHOSETSU PLEIADES
Margaret (Shueisha's manga magazine) → Magnolia
Hashire! Daihatsu Tanto (Japanese-brand car) → Hirameki! Tanto-kun


Source: Japanese Wikipedia, Yahoo! Chiebukuro, ひなぴし - HatenaBlog
P.S. another mind-boggling thing is, One Piece and Naruto were left as is in Bakuman anime.