The intended sense of "structure" comes from Bourbaki, who reformulated all of mathematics as a theory of structures founded on set theory in their multi-volume Elements of Mathematics (with the obvious allusion to Euclid), see Nicolas Bourbaki: Theory of Structures. A structure is a set with operations and/or relations defined on it and/or additional objects attached to it. Topology, metric, group operation, partial order, etc., are examples of structures.
In differential geometry the basic structure is the differential one, and one can put extras on top of it, affine connection, Riemannian metric, symplectic form, complex or Kähler structure, etc. Despite having prominent geometers among its members and decrying the "debauchery of indices" in classical expositions, see What was the motivation for the development of modern, intrinsic, differential geometry?, Bourbaki never quite got to writing its own volume on differential geometry.
Kobayashi-Nomizu's Foundations of Differential Geometry became the stand-in for that. It promoted systematic use of invariant notation and intrinsic descriptions of geometric objects through the systematic use of the language of Lie groups, homogeneous spaces, fiber bundles, connections on bundles, and other differential geometric structures. Nomizu particularly contributed to the revival of interest in affine structures in 1980-s with his Munster lecture What is Affine Differential Geometry? and subsequent work, see Geometry And Topology Of Submanifolds VII, p.35.