Hello, I have a question about the training from Academys as a group patron.

Training. Because you're associated with the academy, you receive a discount on any education you wish to pursue. When you undergo training as a downtime activity (as described in the Player's Handbook or Xanathar's Guide to Everything), you pay half the normal cost, assuming the academy
teaches that subject. Training in languages, musical instruments, and other tools is also available, at the DM's discretion. In addition, you can gain proficiency in the Arcana, History, Nature, or Religion skills by this method, as if you were learning a language. A character can learn only one of these skills in this way.

If a character learned History as the method with a language, the character can do this only once, is it right ?

But the character can learn languages, musical instruments and other tools as described in Xanathar's Guide, is it right ?

Correct "you In addition, you can gain proficiency in the Arcana, History, Nature, or Religion skills by this method, as if you were learning a language. A character can learn only one of these skills in this way." But you only get one of those proficiencies/skills.  For languages any in your can find in PHB or XG. 

Yes. You can do any of the normal training as much as you want, and 1 time you can do language training for 1 of 4 skills.