I am not sure how Kartikeya's relationship is depicted in the North Indian sources, but I can surely tell you how Tamil literature describes the brothers' relationship.

First of all, down South, Kartikeya is the younger brother and Ganesha the elder one. So there is a reversal of roles. OTOH, I can recollect three incidents which shed light on the brothers' relationship.

First is the circumambulation of the Earth- in which Ganesh is declared the winner. In the Tamil sources, this incident takes place when they were still children and the reason for holding the competition was to determine who would get to eat the mango that Narada had brought and offered to Shiva. Kartikeya is rather miffed that his parents' agreed with Ganesh's 'shortcut' parikrama.

 Kartikeya's anger is more against his parents' than his brother. But as far as I have read, it is the sulking and disappointment of a young child- not the vindictiveness of a teenager as being shown in the serial. In fact Parvati goes on to cajole Murugan.

The second incident where we get to see both of them is just before the war with Surapadman when Kartikeya prostrates before Vinayaka and seeks his elder brothers' blessings before starting out (since Ganesha is considered Vignaharta- remover of all obstacles). 

The third and  most famous incident is when Murugan tries to woo Valli but runs into one problem after another. Getting frustrated, Kartikeya mentally invokes Ganesha to come and help him out. 

and Ganesha immediately responds (what are elder brothers for, if not solve their siblings' problems?) by appearing as a huge wild elephant. 

(I had written about Murugan- Valli romance somewhere- not able to locate it right now)

Apart from these Puranic accounts, there are several description by poets and saints describing the quaint, sometimes amusing, activities of Ganesh- Kartik.