Isn't taking away Kogasa's umbrella an equivalent of ripping someone's limb off?
Like BlindVigil said, it's not like a limb, but it is part of her, try seeing it as Banki's heads, you can take them away but it doesn't really hurt her. She does, "feel" it? I suppose, I don't how far can she feel her umbrella, I'm sure that if for example, you were to fight Kogasa, Why would you?, your best bet is attacking the umbrella since it's her main body, but I don't know if she can feel touches or let's give an example, you tickling the umbrella.
Like BlindVigil said, it's not like a limb, but it is part of her, try seeing it as Banki's heads, you can take them away but it doesn't really hurt her. She does, "feel" it? I suppose, I don't how far can she feel her umbrella, I'm sure that if for example, you were to fight Kogasa, Why would you?, your best bet is attacking the umbrella since it's her main body, but I don't know if she can feel touches or let's give an example, you tickling the umbrella.
Umbrellas don't usually have nerves, but to become a youkai they demonstrate they do have feelings, so it's possible they also 'feel' physically; Kogasa is 'an umbrella come to life with a detachable cute girl' more than she's 'a youkai holding an umbrella', after all, so... it might be like a Soul Gem.
But perhaps the girl is the 'gem'?
Anyone's guess whether separating them too far/too long would have consequences though. If Sekibanki can generate extra heads, can Kogasa multiply moéblobs?