I noticed that both 死ぬ and the 音読み of 死 share a し sound. Is this a huge coincidence between Japanese and Chinese, or is there some sort of relation? I guess the former, because I don't know any function ぬ may have after a borrowed noun, but I don't know much about etymology.
It is a tempting identification, but the 〜ぬ suffix is inexplicable. In fact, there are only two n-stem verbs in Old Japanese – 死ぬ and 去【い】ぬ – plus one auxiliary (the perfective 〜ぬ), all of which are conjecturally related. Linguistic coincidences are not unheard of: one well-known example concerns the word "dog" in English and in Mbabaram.
For what it's worth, the 日本語国語大辞典 records the following etymological theories:
By contrast, it is generally accepted that words like 馬【うま】 and 梅【うめ】 are ancient loanwords from Chinese.