Yes, the holographic principle can indeed be extended to a 2-dimensional universe. In fact, the original formulation of the holographic principle, proposed by physicist Gerard 't Hooft and later developed by Leonard Susskind and Juan Maldacena, involves a correspondence between a higher-dimensional space and a lower-dimensional boundary.
According to the holographic principle, a theory in a certain number of dimensions can be mathematically equivalent to a different theory in one fewer dimension. In other words, the physics of a higher-dimensional space can be encoded or "projected" onto a lower-dimensional boundary without any loss of information.
In the specific case you mentioned, a 2-dimensional object can indeed have its information fully encoded on a 1-dimensional boundary line surrounding it. This means that the properties, dynamics, and all relevant information of the 2-dimensional object can be represented on the boundary line, much like a hologram encodes a three-dimensional image on a two-dimensional surface.
This idea of holography has been studied extensively in the context of the AdS/CFT correspondence (Anti-de Sitter/Conformal Field Theory correspondence), which relates a theory of gravity in a higher-dimensional Anti-de Sitter space to a conformal field theory living on its boundary. This duality has provided valuable insights into the nature of quantum gravity and has deepened our understanding of the connections between spacetime, gravity, and quantum field theories.
It's worth noting that the holographic principle and its extensions are still active areas of research, and further exploration is required to fully comprehend their implications and applications in different dimensions and physical contexts.