Kindergartners are generally at an early stage of geometric development, in which shapes are recognized by how well they resemble prototypical images, rather than by whether or not they conform to a definition. Thus, for example, the shape on the left below is likely to be recognized as a "triangle" (despite the fact that it has four sides), the shape in the middle is is unlikely to be recognized as a "triangle" (despite the fact that it has three sides), and the shape on the right is likely to be recognized as a "diamond" (and not as a square, despite the fact that it is a regular quadrilateral). In the Van Hiele model, children at this level would be described as being at the "visualization" level, or Level 0. In order to recognize that a square is a rectangle they would need to be at the "abstraction" level, or Level 2, at which hierarchical relationships can be understood.
I would say that one of the goals of early childhood education (roughly K-2) is to move kids from Level 0 to Level 1, with the transition to Level 2 taking place in later elementary (say grades 3-5 or 6). While certainly some kindergartners are developmentally ready for understanding some hierarchical relationships (e.g. "poodles are dogs, and dogs are mammals"), I think expecting this to be a goal of kindergarten (something they are "supposed to learn") is unrealistic.
Parenthetically, it's probably worth mentioning that the Van Hiele model is widely regarded by math ed researchers as outdated, overly simplistic, and extremely reductionist. However,
much as Euclidean geometry maintains a stable position in the
secondary curriculum despite the efforts of some reformers of the
early and mid-twentieth century to jettison it in favor of more modern
approaches to the field, so too does the van Hiele theory continue to
play a dominant role in the discourse of students’ thinking about
geometry, notwithstanding several decades of critique regarding its
value and validity as an empirically-grounded theory.
(Source: Herbst, Fujita, Halverscheid and Weiss, The Learning and Teaching of Geometry in Secondary Schools, pp. 92).
There are decades and decades worth of research on these matter, but an accessible point of entry might be:
Hannibal, M. A. (1999). Young children's developing understanding of
geometric shapes. Teaching Children Mathematics, 5(6), 353+.
In fact, the entire issue of Teaching Children Mathematics in which that article was published may also be of use, as it was a themed issue on geometry.
