Both Gautama (patronymic /descendant of the Vedic Ṛṣi Gotama i.e. born in his family/gotra) & Āṅgirasa (patronymic /descendant of the Vedic Ṛṣi Aṅgiras i.e. born in his family/gotra) are patronymics & serve as brahmin family names. Every brahmin in the clan of the Ṛṣi Gotama would automatically also be an Āṅgirasa (as the Vedic Ṛṣi Gotama was himself a descendant of the Vedic Ṛṣi Aṅgiras but established his own clan having personally seen 7 generations of his own lineage i.e. ancestors/descendants and thereby becoming eligible to establish his own gotra). Similarly the Bhāradvājas are another gotra (descendants of the Vedic Ṛṣi Bharadvāja), and Bharadvāja himself was also a descendant of the Vedic Ṛṣi Aṅgiras - thus all Bhāradvājas are also ipso-facto Āṅgirasas.
Each gotra is exogamous so no Gautama-gotra man would normally marry a Gautama-gotra woman (by marriage the wife moves into the husband’s gotra). So the Buddha’s family would all be gautamas (just as he himself was, his mother becoming a gautamī i.e. wife of a gautama by marriage) but not everyone in the Buddha’s place of birth would be from the same gotra as that would have made marriage impossible and the need for gotra names meaningless. The claim of belonging to Āditya-gotra (or claiming oneself to be a kinsman of the sun i.e. āditya-bandhu) is simply a boast - a boast that the Ṛṣi Yājñavalkya also used from time to time to refer to himself as āditya-bandhu (and the Śukla Yajurveda tradition says Yājñavalkya, who is regarded as that tradition’s progenitor, got the veda from the sun himself). Yājñavalkya was a brahmin from Kosala as well, as recorded in the Śatapatha Brāhmaṇa (a text whose last part forms the Bṛhadāraṇyaka Upaniṣad). There were Gautama families (i.e. belonging to the gotra of the ancient Ṛṣi Gotama) living in Kosala in the time of Yājñavalkya. The brahmin Pauṣkarasādi (called Pokkharasāti in the Pāli canon) was an Aupamanyava (a descendant of Upamanyu, who belonged to the Gautama gotra) and Pauṣkarasādi was himself a Gautama. He may have supported the historical Buddha (a Gautama) perceiving him as his close/distant kinsman.
I believe (based on the name of the Kapilasuttam in the Sutta Nipāta, which is also alternatively titled dhammacariyasuttam) that the Buddha was originally named Kapila (and his birth place got retrospectively named Kapila or Kapila-vāstu (i.e. Kapila’s home) after him. Kapila (being variant name of Kapilavāstu) is found in the Nettippakaraṇa (“kapilaṃ nāma nagaraṃ suvibhattaṃ mahāpathaṃ”) and in the Therāpādāna (anukampiya sakyānaṃ upesi kapilavhayaṃ). He may not have been elsewhere called by his proper name in the canon as it was considered rude back then to refer to one’s teacher by their given name, so he is normally called bhagavān by his disciples and by his gotra name by others.