How does Shingo's use of "osu" in this particular dialogue differ from its use elsewhere? In the manga Osu!! Karatebu the word "osu" is used quite frequently, and from context I have so far assumed that it is a way for the junior members of the karate club to respond to their seniors in a general "yes, sir" way. However, I recently came across this panel which presents a quite different take on the word:



What is the significance of Shingo's particular dialogue about "osu?" Is it a typical philosophy of the character, karate, or a mix of the two? Why is it so different from how it is used elsewhere in the manga?
You make an overly broad assumption with this particular word. "Ossu" here is talked about not as a salutation here, but for a deeper more philosophical meaning in general.

Ossu, is made of of two kanji:

[押]{おっ} and [忍]{す}. 押 by itself loosely translates to the action of apply pressure to something, such as pushing. 忍 means endurance or restraint.

The dialogue by the character here breaks down the literal meaning of the word and uses the two kanji as a philosophy of sorts. It's like using the word embodiment of a word as a sort of ambition or goal to live by. 

In this case the spirit of Ossu is made up of these two kanji. The meaning here is to apply them meaning of those to your own life and when you use them remember their meaning so that it may carry you further, whenever or wherever you may need it, like a mantra.