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hm, I don't think so - non-Newtonian fluids are like water + cornstarch: they flow like a liquid unless you apply force, then they harden into a solid.
The tomato paste is just not a liquid, it's so thick it functions like a solid. So if you spoon some of it into a jar, the paste will just stay where you spooned it, it won't settle like a liquid would. AFAIK, a non-Newtonian fluid would settle like a liquid when not under force, unlike the tomato paste.
Not all non-Newtonian fluids are like this. Some are reversed: They flow like a liquid when pressure is applied, and harden into a solid (or gel) when the pressure is released. I think ketchup is one such example.
https://en.wikipedia.org/wiki/Quasi-solid