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The moment had confused many fans when it aired last month, since previous incarnations showed The Doctor sporting their predecessor's clothes for an episode following their regeneration.
Doctor Who has regenerated to a subscription platform after a major deal between BBC and Disney+.
The actress, who is also the daughter of the Fifth Doctor Peter Davison, shared a cute black-and-white snap with David in matching hats, after his return to Doctor Who aired on Sunday.
The Sex Education star, 30, took part in an Instagram Q&A on Tuesday to speak about the 'big shoes' he will fill and revealed he'd love to fly the time traveller's police-box to Nigeria.
The broadcaster will air a week of special programming as part of its centenary celebrations.