That's an unwarranted assumption, IMHO, if it predates the divergence between the plant and animal kingdoms.
How does a gene that build fungal cell walls, learn how to build blood vessels? (completely different structures)
I'm not so sure. Both spheroidal and cylindrical objects require rounded surfaces.
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A really great book showing this kind of species transfer/economy of design is “Endless Forms Most Beautiful: The New Science of Evo Devo”, by Sean B. Carroll, 2005. For example the earthworm’s segments, trilobite segments, insect segments and mammalian spine are developmentally related.