Simple Region plot with "or" condition (||) does not connect the or parts "smoothly"?

It looks like the default discretization method gives the desired singular vertex:

reg = ImplicitRegion[x <= y && x < 2 || x >= y && x >= 2, {{x, 0, 5}, {y, 0, 5}}];
mesh = BoundaryDiscretizeRegion[reg, Frame -> True]

We can verify this by highlighting the vertices:

HighlightMesh[mesh, Style[0, Red]]

See Specific initial sample points for 3D plots for some discussion of following form of PlotPoints. Basically throw a point into the offending area and increase MaxRecursion (default value is 2), if needed. If you follow the linked Q&A, you'll realize this approach has a good chance to succeed and a still significant chance that it will fail. It succeeds here if we raise MaxRecursion to 3:

RegionPlot[x <= y && x < 2 || x >= y && x >= 2, {x, 0, 5}, {y, 0, 5}, 
 MaxRecursion -> 3, PlotPoints -> {Automatic, {{2.001, 2.01}}}]