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As for the written characters, as both words are pronounced very similarly it makes sense that they share a phonetic component. But where do the spoken words actually come from? It seems very likely to me that there is a connection between the noun "bag" and the verb "to carry", and that they should be pronounced the same except for their tone would suggest that there is. Does anybody know?
Interesting parallel with 带着的带 and 袋子的袋...
Regardless of whether you want to talk about "etymology" or "glyph origin", the first step is to find the earliest appearances of the character and try to gauge what kind of word or words it represented. Bag is not the earliest meaning of 「包」; the word represented by this character should be interpreted as either
野有死麕、白茅「包」之
In the wild there is a dead antelope,
And it is wrapped up with the white grass.
令汝辟百僚，有尚事「包」，廼多亂。
I order you to oversee and govern the officials. If bribery occurs, chaos will ensue.
Note: "Bribery" is a metaphorical translation of 「包」, which should be interpreted as wrapped gifts in this context.
Baxter-Sagart (2014)'s reconstructions and explanations may offer some insight:
Here, the dash "-" indicates a morpheme boundary. The authors list several affixation categories, including an *m- prefix and *<r> infix, which may both occur in related words, e.g.
From this, I would consider it quite likely that 「包」 shares etymology with 「抱」.
包（bāo），现代汉语通用规范一级字（常用字）。 [5] 最早见于甲骨文，像腹中有子形
包 originally meant 'to envelope' using it as a noun for 'bundle' was a natural development of language
包(v)一個包(n) = wrap a wrap (bundle; bun)
抱(hold; embrace) is a 形聲字 (pictophonetic character) coined using the existing 包
you use 手(扌) to hold/embrace, therefore, 扌 was chosen as radical of 抱
包(bāo) is the phonetic element of 抱(bào), since hold and embrace share the sentiment of 'envelope' with 'wrap', thus 包 also provides meaning along with its sound to the word 抱
for this pair, it's a very likely yes.
and it's quite a large topic in Chinese etymology. related concepts include: 分化字, 同源字, in contrast to generally 形声字 where the meaning is not (necessarily) related to the phonetic part.
