In Surat al-Ghashiyah (88:5) Aniyah is used to mean "pinnacle" (of heat). In other words, it is saying that the spring of water has reached its full potential of heat and can't get any hotter. Aniyah itself doesn't mean heat or spring, it just means "full potential" in this verse, and from the context it is known that God means heat.
This same word in Surat al-Ghashiyah 88:5, when used as a name, means a woman who has reached her full potential, who is mature and intelligent.
The "caring person" meaning is for another word, `aniyah, which starts with "ain", not "alif". In English both are spelled and pronounced the same, for this reason you can use the name Aniyah and consider it to mean "mature" and "caring" at the same time.
Another meaning of Aniyah is "vase" or "vessel", a container for drinks, and this meaning is mentioned in Surat al-Insan 76:15 when speaking about Paradise: "Vessels of silver and cups of pure crystal will be passed round among them."
Aniyah with ain is an Arabic word and it is derived from a Quranic root, Ain-W-N (help/refuge/care), and it means a woman who provides help, refuge, or care and thus is caring toward others. In Surat al-Fatihah when we recite "iyyaka na'budu wa iyyaka nastaeen" (Surat al-Fatihah 1:5), "nastaeen" means "we ask for help/refuge/care" [from You O Allah], and it comes from the Ain-W-N root as well.
You can use Aniyah with any of the meanings mentioned above since they are all acceptable meanings.