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In Chapter 3 of Chris Schommer-Pries's PhD thesis, surfaces with corners (more generally, manifolds with faces) are equipped with extra structure called a "halation", with precisely the goal of making the gluing operation a pushout. The topic of making the gluing operation into a pushout in the smooth category is discussed (and motivated) in some detail, quite lucidly.

Added: Alternatively, you could work in the microlinear category, where the problems we are discussing do not arise, and the naive gluing operation is a pushout. See e.g Basic Concepts of Synthetic Differential Geometry by Rene Lavendhomme.