Well, it could be, but you could replace 盛 with practically anything else and it would still function. At least, that’s how it works in Mandarin. 小、中、大. Small, medium, large. I don’t know if Japanese handles them in the same way, but these ‘size’ kanji just refer to a particular size within a given category or type. As 大辞林 puts it in one of だい’s definitions:
③ 大小があるもののうち,大きいほうのもの。「生ビールの―」
Uh… I think that, especially in the case of Japanese, which adopted kanji independently later on… it’s more like… words exist. Some words can be represented by single kanji. Some words are represented by multiple kanji even though they express a single concept, like 今日 (one word, two kanji). At the same time, because kanji are such a common way of representing words now, sometimes, things go in reverse: kanji can be combined to form new concepts, and each kanji’s reading becomes part of the new word.
To put it another way, especially in the case of Japanese… kanji and words exist separately. They just happen to be associated, usually on the basis of meaning, and sometimes on the basis of sound.
I’d say it’s more like… 山 is a picture/diagram/symbol for ‘mountain’ as a concept. It can also be used to represent ‘mountain’ as a standalone word, and in that case, it’s read やま. やま is the word and sound, 山 is the symbol, and ‘mountain’ is the concept and meaning.
Separately, kanji themselves (in Japanese, at least) don’t have grammatical categories of their own. They just represent meanings, and occasionally specific sounds based on common pronunciations.