Technically, you could say the usage takes the meaning because of its etymology.
Vegetate inherits a meaning of passive existence from Aristotelian philosophy translated through the medieval scholastic tradition into the Middle English adjective vegetatif. In early modern English, the verb vegetate emerges referring to growth, and by the 18th century vegetate adopts its modern, figurative sense of passive existence.
Philosophy and Theology
In Aristotle's De Anima, Aristotle specifies three souls (Wikipedia):
- vegetative soul - growth, reproduction
- sensitive soul - mobility, sensation
- rational soul - thought, reflection
Plants are considered only vegetative because they were thought to be incapable of movement, sensation, or thought. Animals are considered to have vegetative and sensitive souls, incapable only of intellectual work. Humans are considered to have all three souls. So, conceptually, vegetative means growth and is also associated with the absence of movement or thought.
De Anima enters the medieval scholastic tradition via a long path through Arabic, including a commentary by Avicenna in the early 1000s and a translation into Latin by Michael Scot in the early 1200s (Wikipedia). Bartholomeus Anglicus uses the term in his De Proprietatibus Rerum (c.1240; see next section for how it's used). A commentary by Thomas Aquinas written near the end of his life (1270s) probably relied on a new Greek translation of the text by William of Moerbeke (Wikipedia). In Aquinas's commentary, the etymological connection to modern English is already clear: anima vegetabilis (vegetative soul). The vegetative soul is one that promotes growth; it cannot by itself produce movement or thought.
The Adjective Vegetative in Middle English
Early Middle English sources draw on the distinctions developed by Aristotle (via the scholastic tradition) to describe what is growing but inert. See these examples from the Middle English Dictionary, vegetatif:
(a1398) Trev.Barth.(Add 27944:Seymour) [John Trevisa's translation of Bartholomaeus Anglicus, De Proprietatibus Rerum (c.1240)] 102/30 : In diuers bodies beþ þre maner soules…vegetabilis þat ʒif lif and no felinge as in plauntis and rotis, sensibilis þat ʒeueþ lif and felinge and noʒt resoun in vnskilful bestes, racionalis þat ʒeueþ lif, felinge, and resoun in men…þe vegetatiue [L anima vegetabilis] desireþ to be, þe sensibil desireþ to be wel and þe resonabil soule desireþ to be best.
[Modern spelling: In diverse bodies be three manner of souls [...]: vegetabilis, that gives life and no feeling as in plants and roots; sensibilis, that gives live and feeling and not reason in unskilful beasts; rationalis, that gives life, feeling, and reason in men. [...] The vegetative desires to be, the sensible desires to be well, and the reasonable soul desires to be best.]
Vegetatif means two things: growth and stasis. Your def. 1, "passive existence without exertion of body or mind" is already consistent with this adjective form, entwined with the sense of def. 2a, to grow. Both meanings started together.
The Verb Vegetate in Early Modern English
Vegetate, as a verb, emerges in early modern English (OED, "vegetate, v."), with the earliest citations around 1600. Its figurative use, now applied to people who are static (def. 6a) emerges by 1740. Note how the two quotes both contrast vegetate and live, as a meaningful active life can't be had while vegetating:
1740 C. Cibber Apol. Life C. Cibber i. 15 The Man who chuses never to laugh..seems to me only in the quiet State of a green Tree; he vegetates, 'tis true, but shall we say he lives?
1777 G. Forster Voy. round World I. 542 In short, we rather vegetated than lived.
By this point the Aristotelian meaning is still around, as is the growth meaning, but the figurative passive meaning has developed as a distinct usage.