Which Bianchi identity is due to Bianchi (or not, since it might be due to Ricci (according to Levi-Civita (according to MO))) or vice versa? wikipedia doesn't say, nor my Berger Panorama book (but I might google Levi-Civita to get rid of one level of brackets) and the library is far (actually not, but it has German Schließungszeiten and I got no key). (I guess the differential Bianchi identity is not due to Bianchi? So who did which?)
To clarify on the "vice versa": according to my Swiss cheese memory, there might be something completely different actually due to Bianchi. Some book was telling this quite nice...
Said MO answer is here
 A: L.P. Eisenhart claims in his Riemannian Geometry (Princeton University Press, 1926, p. 82) that Bianchi was the first who discovered the algebraic identity in 1902.
It is also indicated in Italian Mathematics Between the Two World Wars by Guerraggio and Nastasi (Birkhäuser Verlag, 2006, p. 15)  that Bianchi obtained both of the identities 
which were published in his paper "Sui simboli di Riemann a quattro indici e sulla curvatura di Riemann", Rend. Acc. Lincei, 11 (1902), pp. 3–7.
A: 1.)
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)
2.)
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)
A: Wikipedia saids that the algebraic one at least is due to Ricci:
Curvature of Riemannian manifolds - Symmetries and identities
A: The first one is due to Ricci, and the second one is due to Bianchi.
