I am learning Mandarin using Lingodeer and I've encountered the sentence, “这个女孩眼睛大” and I am wondering why it isn't necessary to refer to the girl's eyes with “的” in this sentence? I know that “的” can be left out when referring to someone you're intimate with or related to but I don't think your eyes fall into either of those categories...?
In 这个女孩眼睛大, 这个女孩 (this girl) is the subject and 眼睛大 is an adjectival phrase for it. Meaning "[这个眼睛大的][女孩]" (this girl with big eyes)
In 这个女孩的眼睛大, 女孩的眼睛 (this girl's eyes) is the subject and 大 is an adjective for it. Meaning "[这个女孩的眼睛][很大]" (this girl's eyes are big)
[他][眼大] = [he] [is big-eyed]
[他的眼][大] = [his eyes] [are big]
[他][手快] = [he] [is fast-handed]
[他的手][快] = [his hands] [are fast]
It is actually related to the sense you describe, but not exactly the same. You should think of dropping "的" as implying that there is an intimate connection, in a general sense and not specifically "romance", between the subject and the possessed.
You are intimately possessed of your eyes as they are a part of your body, hence, you can drop "的".
That said, it is not wrong to use "的" in that you won't be understood if you do, but it does add a more formalizing air.