The portion you quoted in your question is from Kathasaritsagara and the exact verse number is 92. In that verse the names of two sons of Kartikeya are mentioned. For the start let's take a look at original sanskrit shloka which is not present in English version of your post.
तद्वज्राभिहतस्याङ्गात्षन्मुखस्योद्वभूवतु: |
पुत्रौ शाखविशाख्याख्यावुभावतुलतेजसौ ||92 ||
When Indra heard that, he was troubled, considering that his kingdom
was taken from
him, and being jealous he went and made war upon Kārttikeya. But from
the body of Kārttikeya, when struck by the thunderbolt of Indra, there
sprang two sons called Śākha and Viśākha, both of incomparable might.
Here we can clearly see the word "पुत्रौ" or sons is been used by the author.That means Śākha and Viśākha are called as sons of Kartikeya.
As you have enquired Sakha , Visakha called as the sons of Skanda are mentiond in various Hinduism scriptures like Mahabhrata and Puranas etc.
But unlike Kathasaritsagara they are not called as sons of Karttikeya but the incarnations , amsha (part) , brothers and followers along with Naigameya and Astavan , the other two. Now let's take a look at what is the relation according to scriptures.
Mahabharata in it's Shalya Parva-Gada Parva- Chapter 44 is mentioning their names in the topic related to birth of Kartikeya.
There is a story in that chapter in which Lord Shiva ,Devi Uma , Ganga and Agnideva was curious to see to whom amongst the four the Karttikeya would first approach for honouring .Then the Kartikeya Understanding the expectation of these four assumed four different forms at the same time VIZ. Sakha , Visakha and Naigamey. Among the four forms Skanda went to adore Rudra or Shiva. Visakha went Uma or Parvati. Sakha, which is Kartikeya's Vayu form proceeded towards Agni. Naigameya proceeded to the presence of Ganga.
So this is the story found in Mahabharata about them. But here they are called as separate forms of Karttikeya with similar appearance.
ततोSभवच्चतुर्मुर्ति: क्षणेन भगवान प्रभु: |
तस्य शाखो विशाखस्य
नैगमेयश्च पृष्ठत : ||
he had recourse to his Yoga powers and assumed at the same time four
different forms. Indeed the adorable and puissant lord assumed those
four forms in an instant. The three forms that stood behind were Sakha
and Visakha and Naigameya. The adorable and puissant one, having
divided his self into four forms, (proceeded towards the four that sat
expecting him). The form called Skanda of wonderful appearance
proceeded to the spot where Rudra was sitting. Visakha went to the
spot where the divine daughter of Himavat was. The adorable Sakha,
which is Kartikeya's Vayu form proceeded towards Agni. Naigameya, that
child of fiery splendour, proceeded to the presence of Ganga. All
those forms, of similar appearance, were endued with great effulgence.
The four forms proceeded calmly to the four gods and goddesses
(already mentioned). All this seemed exceedingly wonderful.
Now lookig at Puranic versions where these names are found , we can see that they are refereed here as his younger brothers , incarnations and Amsha. Here we can yet another form called as Astavan.
Matsya Purana in it's chapter No.5 states that Sakha ,Visakha and Naigameya were Skanda's younger brothers.
अग्निपुत्र: कुमारस्तु शरस्तम्बे व्यजायत |
तस्य शाखो विशाखस्य
नैगमेयश्च पृष्ठजा : ||26 ||
Kumara the son of Agni was born in the forest of reeds.His younger
brothers were Sakha , Visakha and Naigameya were well Known
Here is Hindi translation of above verse by Gita Press.
Matsya Purana at Part 2- Chapter -159- Verse 3 calls Skanda by the name of Visakha .
यतस्ततो विशाखोSसौ ख्यातो लोकेषु षण्मुख: |
स्कन्दोविशाख: षड्वक्त्र:
कर्तिकेयस्च विश्रुत:||3||
The krttikas joined together with him after his birth and the six
heads were fixed to the six trunks ; for these reasons he is known as
skanda , Visakha , Sanmukha and karttikeya.
Here is Hindi translation of above verse by Gita Press.
Vayu Purana Chapter 66 - Verse 24 , joining in the league of puranas also mention these as younger brothers of Kumara. Note here that Vayu purana verse is exactly similar as of Matsya purana.
स्कन्द: सनत्कुमारश्च जज्ञे पादेन तेजस : |
अग्निपुत्र: कुमारस्तु
शरस्तम्बे व्यजायत ||
तस्य शाखो विशाखश्च नैगमेयश्च पृष्ठजा : ||24
||
skanda and Sanatkumaras were born of the fourth part of Tejas (fire)
of Agni , the son Kumara was born in the sara grove (a kind of reed) .
Sakha , Visakha and Naigameya were his younger brothers.
Similarely in the same purana Chapter 101- verse 280 states that these are incarnations of Skanda and are anuchara अनुचर or followers of him. In this verse a another name Astavan is also mentioned.
तस्य शाखो विशाखश्च नैगमेयश्च चाष्टवान् |
व्यपेतवयसना: क्रुरा:
प्रजानां पालने रता: ||280 ||
He has four followers , VIZ. Sakha , Visakha , Niagameya and Astavan
(these are incarnations of Skanda). They are devoid of vicious
indulgence , uncruel ones and engaged in protection of progenies or
subjects.
Conclusion - So according to Hinduism Scriptures these are brothers of Skanda also calling them part , companions , followers and incarnations of him.
It is quite possible that the author is calling these as sons of Skanda because many a times it is seen in Hinduism scriptures that sons are also sometimes referred as Amsha अंश. So from that perspective the author has used the word son Amsha , where as these are the parts , incarnations , brothers of Skanda. But we can infer that although they are called differently the central meaning of theses is altogether the same and the meaning of the words Brother , Amsha , followers etc. are interchangeable. They are called as his sons poetically.
