Barack Obama did not serve a third presidential term under Joe Biden and Kamala Harris; the phrase 'Obama's third term' is used metaphorically by some commentators to describe policy continuities or influence, not an actual third term in office.
Obama Served A third Term Under Biden/Kamala.

By Trump's flawed logic, even Obama can seek a ‘third term’: Will 12th amendment stop it? - Hindustan Times
Source: hindustantimes.com
Key Evidence
The 22nd Amendment limits U.S. Presidents to two terms; Obama served two full terms from 2009-2017.
hindustantimes.comhindustantimes.comBy Trump's flawed logic, even Obama can seek a ‘third term’: Will 12th amendment...
Multiple opinion pieces use 'Obama's third term' metaphorically to describe Biden's policy continuities with Obama's administration.
manhattan.institutemanhattan.instituteBiden Is Delivering Obama’s Third Term
spectator.co.ukspectator.co.ukWe’re living through Barack Obama’s third term
No credible source provides evidence that Obama holds any official presidential role during Biden's administration (all sources).
What the Evidence Shows
Legally, Barack Obama served two terms as President of the United States from 2009 to 2017, which is the maximum allowed under the 22nd Amendment to the U.S. Constitution. There is no legal or factual basis for Obama serving a third term.
Multiple sources use the phrase "Obama's third term" metaphorically or rhetorically to describe the Biden administration's continuation of policies or political style associated with Obama. For example, (Manhattan Institute) discusses Biden delivering policies reminiscent of Obama's tenure, such as extended unemployment insurance.manhattan.institutemanhattan.instituteBiden Is Delivering Obama’s Third Term
(The Spectator) and (Forbidden Knowledge TV) reflect on perceptions that Obama’s influence remains strong in the current administration, sometimes suggesting he is effectively running the show behind the scenes. However, these are opinion-based or speculative narratives rather than statements of fact.spectator.co.ukspectator.co.ukWe’re living through Barack Obama’s third term
forbiddenknowledgetv.netforbiddenknowledgetv.netWe’re in Obama’s Third Term
(Political Moonshine) and (American Thinker) discuss conspiracy-like or critical views framing Biden’s presidency as a continuation of Obama’s legacy, but do not provide evidence that Obama holds office or official power.americanthinker.comamericanthinker.comObama's Third Term and the Destruction of the American Polity
politicalmoonshine.compoliticalmoonshine.comObama’s Third Term
(Hindustan Times) clarifies that constitutional rules prevent any president from serving more than two terms, and even discussions about loopholes for other presidents do not apply to Obama.hindustantimes.comhindustantimes.comBy Trump's flawed logic, even Obama can seek a ‘third term’: Will 12th amendment...
(Reddit discussion) explicitly notes that Obama has already served two terms and cannot run again, reinforcing the constitutional limit.reddit.comreddit.comr/changemyview on Reddit: CMV: Obama needs to hit the campaign trail until Trump...
Overall, while commentators and political analysts sometimes refer to Biden’s presidency as "Obama’s third term" to highlight policy continuity or influence, this is a figurative expression rather than a factual claim about Obama holding office again.
Therefore, the claim that "Obama served a third term under Biden/Kamala" is false when interpreted literally as Obama holding presidential office during Biden’s administration.