What we call potato is "the battery" of the plant.
The starch(?) is the energy source for re-growing after winter.
Well, that sound a bit simplistic. Besides having potential energy in the form of matter (starch) that can be oxidized, if oxygen is available as well (batteries have everything they need to produce energy right in them, they are self-contained energy producers -- a potato, requiring that oxygen be mixed with it before energy can be produced, might be better analogized as being the gasoline of the plant, than as the battery.
Potatoes, as you indicated, also serve as sources of cells for vegetative reproduction -- cloning of a new plant after the winter. I can't think of a mechanistic analogy for vegetative reproduction -- reproduction is one of the things that distinguishes the living from the non-living: I've never heard of a battery that makes battery factories that then make more batteries.
By the way, if you were really strictly vegan, you might want to avoid eating peas. Pea plants are carnivorous predators. Sprouting pea-seeds exude a sticky substance that traps insects that live in the soil, which results in the insects' early demise, and which enables the growing pea plant to "eat" them. Pea plants have very high nitrogen requirements, and decaying animals produce more nitrogen than decaying plants. If baby pea plants didn't kill insects, they might not survive long enough to become young-child pea plants. By choosing to eat a pea instead of a potato, you are choosing to kill more insects. Soil-crawling insects are the "hidden animal ingredients" of pea plants.
Of course once the pea root system reaches a certain size, and develops a greatly increased surface area for absorbtion of soil nitrates, it no longer needs to kill insects to get enough nitrogen. But if it wants to live long enough to develop an adequate root system, it may have to depend on it ability to kill insects, until that time.