You misunderstand 分 here.
分 does not mean "division" here. Here 分 means the "points" (test score), in an exam or a contest where the perfect score is ten.
分 can also means "perdec" (per ten, a tenth), as a synonym of 成.
三分 three points (out of ten), three perdec (3/10)
五分 five points (out of ten), five perdec (5/10)
十分 ①[original meaning] ten points (out of ten), perfect score, full marks ②[extended meaning] "perfectly", very, extremely
幾分 how many points (out of ten), how many perdec
Assuming that there is an exam called "love", how many points can I get from the exam?
我愛你有幾分?
How many points (out of ten) do I love you?
How many perdec do I love you?
= How much do I love you?
Extra examples:
三分天注定,七分靠打拼。 [famous lyrics]
Three points depend on fate, while seven points depend on hardworking.
3/10 depends on fate, 7/10 depends on hardworking.
[Chinese version of "nature versus nurture"]
逢人只說三分話,未可全拋一片心。[ancient proverb]
Every time you meet a person, you should only tell them three perdec (3/10) of your mind. You should not "throw the entire heart to them" (tell them all your mind).
Disputes
As 分 is also a traditional Chinese unit of length, which is modernlycurrently defined as ⅓ centimetre (in mainland China) or 10/33 centimetre (in Taiwan), some people believe that it is better to parse it as the depth of love measured in this length unit, instead of points or perdecs.
Examples:
入木三分 [idiom]
[literally] into the wood for three fen of depth
[figuratively] written in a forceful hand, penetrating, profound
Some knowledge about the Chinese culture of proportions
Western culture prefers to use "percent" (per hundred, a hundredth), while Chinese culture traditionally uses "perdec" (per ten, a tenth), known as 分, 成, 折.
Examples:
三分 = three perdec (three tenths, 3/10)
七分 = seven perdec (seven tenths, 7/10)
八成 = eight perdec (eight tenths, 8/10)
九成 = nine perdec (nine tenths, 9/10)
折 is only used for discounts.
九折 = nine perdec remaining (i.e. 10% discount)
八折 = eight perdec remaining (i.e. 20% discount)
八五折 (8.5折) = 8.5 perdec remaining (i.e. 15% discount)