According to Wikipedia, the tesseract is the four-dimensional analogue of the cube and has the Schläfli symbol {4,3,3}, and Wikipedia features the following visualization:
However, when looking at it, I wonder whether this could also be a stereographic projection of a tessellation of spherical 3-space (the analogue of spherical geometry in three dimensions) using three cubes around every edge, from spherical 3-space to Euclidean 3-space (except the fact that the edges are straight in this image, whereas they would be bent in the stereographic projection), similarly to what the icosahedral honeycomb {3,5,3} (which is a tiling of hyperbolic 3-space using the icosahedron) looks like in the Poincaré ball model:
So, my question is, does the Schläfli symbol {4,3,3} unambiguously represent the tesseract, or could it also represent a cubic tessellation of spherical 3-space that has three cubes around every edge? If it cannot represent this tessellation of spherical 3-space, does the tessellation have some other Schläfli symbol (in that case, what?), or can't spherical 3-space be tessellated in this way at all?
Schläfli symbol only captures topology. You are right that this Schläfli symbol represents a cubic tessellation of spherical 3-space, but topologically there is no ambiguity here: the tesseract is that tessellation, just as the cube is a square tessellation of the sphere, or the square is a tesselation of the circle. (Yes, they are all finite.)
If you want to distinguish between tessellations of the $(n-1)$-sphere and polytopes in the $n$-dimensional Euclidean space, you indeed have to specify this underlying geometry "manually".