What is commonly drawn as a tesseract is a 3D projection of a 4D cube, just like you can draw 3D objects on a 2D piece of paper with perspective.
In physics spacetime is modelled as a 4D space, but the notion of distance in this 4D space is not the same as the notion of distance used to build the tesseract or normal 4D cube. To build the latter all dimensions are on equal footing and adjacent points are all at the same distance, the distance in this 4D space is just the natural generalization of distance between two points in 3D space. In 4D spacetime the notion of distance is strange (the so called Lorentzian metric) which gives rise to all sorts of weird stuff like worldlines.
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