There has been some discussion about the name Gotama, and how a Brahmanical clan name became associated with a khattiya family. It has been proposed that Gotama was a personal rather than clan name.
However, not just the Buddha, but his step-mother, and Ānanda are called Gotama; decisively, even the assembled Sakyans are addressed by the Buddha as “Gotamas” (gotamā) at SN 35:243:2.8. This rules out the “personal name” thesis.
No-one seems surprised that the Buddha’s clan name was Gotama. In fact it was common for khattiyans to be referred to by brahmanical priestly names. The Mallians, khattiya members of the Vajji league, are referred to as Vāseṭṭhas (dn16:5.19.2, dn33:1.4.5). Several people are referred to as Aggivessana (Aciravata at mn125:2.4, Saccaka at mn35:4.2 and mn36:5.1, and Dīghanakha at mn74:2.5) and while their caste is not explicitly stated it is likely they were khattiyas.
It was pointed out long ago by Chalmers that the “Aggivessanas” were so-called after their brahmanical family priest. This custom is discussed as part of the Pravara ceremony in the Aitareya Brahmana 7.25, of which you can see a translation here and the Sanskrit text here.
tasmāt tasya purohitasyārṣeyeṇa dīkṣām āvedayeyuḥ
Therefore they proclaim him as an initiate with the family priest’s lineage
It seems that when a khattiya was initiated, he was temporarily made into a brahmin when he covered himself with a black goatskin, and his status as a khattiya was restored at the end of the rite. But his family would continue to use the name of the priest as a kind of honorary clan name.
Chalmers’ conclusion was endorsed by Malalasekera in his Dictionary of Pali Proper Names, where he notes under Aggivessana:
Probably the name of a brahmin clan, the Agnivesyāyanas, and the Ksatriyas who were so styled, took the name from their brahmin purohitas
Yogendra Mishra (An Early History of Vaishali, pg. 110) notes that the Licchavis are called Vāseṭṭhas in the Jain literature, and says:
We know from the Aitareya Brahmana that the gotra or pravara of a Ksatriya is the same as that of his Purohita or family priest, who makes him perform the sacrifices. The Vāsiṣṭha gotra was therefore the gotra of their family priest, and we know that the Vāsiṣṭhas were the family priests of the kings of the solar race, especially that of the Ikṣvākus.
The latter statement is interesting, for the Buddha was of the the Ikṣvākus, yet his family priest was Gotama, which suggests the above rule was not universal.
The Buddha is also commonly referred to in devotional verse as Aṅgīrasa, which is the name of one of the ancient Brahmanical rishis (an5.195:3.3, thag21.1:44.3, thag10.1:10.1, dn32:3.14, sn8.11:3.3, sn3.12:11.3).
Malalasekera says that:
Gotama and Angiras are both enumerated in the Pravara ceremony as the ancestors of the Gotama clan
The Pravara ceremony is the same initiation as referred to above. Here’s the Sanskrit.
gotamānām.āṅgirasa.āyāsya.gautama.iti./.ucathyānām.āṅgirasa.aucathya.gautama.iti./.rahūgaṇānām.āṅgirasa.rāhūgaṇya.gautama.iti./ (pravara: Aṅgirases) (AsvSS_12.11-1a)
soma.rājakīnām.āṅgirasa.sauma.rājya.gautama.iti./.vāmadevānām.āṅgirasa.vāmadevya.gautama.iti./.bṛhadukthānām.āṅgirasa.bārhaduktha.gautama.iti./.pṛṣadaśvānām.āṅgirasa.pārṣadaśva.vairūpa.iti./ (pravara: Aṅgirases) (AsvSS_12.11-1b)
Thus it was an accepted custom for khattiyas to be referred to by the name of their Brahmanical family priest. Gotama, Aggivessana, and Vāseṭṭha are all used in this way. There are surely other examples, but this is enough to establish that this is a normal usage and requires no special explanation in the case of the Buddha.