The algebraic Bianchi identity must have been known to Riemann, as shown by his posthumously published prize essay to the Paris academy, "Commentatio mathematica ...". Cf. Spivak Vol. II which has an english translation and modern interpretation. (Interesting for German readers: Anmerkungen by Dedekind and Weber in "Bernhard Riemann's gesammelte Mathematische Werke" 1876, 2.Aufl. 1892, p.405ff.)
First explicit public appearance with proof is in:
E.B. Christoffel: "Über die Transformation der homogenen Differentialausdrücke zweiten Grades", J. reine angew. Math 70 (1869) 46-70
Formula $(16^d)$ on p.55 (via Göttinger Digitalisierungszentrum)
The differential Bianchi identity is due to Ricci. Paraphrasing a footnote in Levi-Civita's "The Absolute Tensor Calculus" (1928/1947) p.182: It was first published without proof by Padova (*) 1889 on the strength of a verbal communication by Ricci. Then it was forgotten even by Ricci himself. Bianchi rediscovered it and published a proof in 1902.
(*) E.Padova: "Sulle deformazioni infinitesime: nota", Atti della Reale Accademia dei Lincei, Rendiconti (1889) Serie 4, Volume 5, 1° Semestre, 174-178
See p.176 footnote (1) (via archive.org)