I won't offer advice regarding what kind of pet to get. But I will say that YES it is possible to "become" a dog or cat person. Much like a person who isn't much into babies becomes a fantastic mommy/daddy once his/her own child is born.
If someone else has a pet and foists it upon you (watching your neighbor pick up poo, is sort of the same concept) then it is much different than YOU choosing to take an animal into your life. You will spend much time with it and it will love and trust you for everything and you will bond with it as well. UNLESS you get a pup and throw it into the back yard. If that is an outside plan... don't do it. It's an insidiously vicious cycle that keeps you from spending time with the dog. You put it out b/c of the mess, or when you're not home, and it gets dirty/smelly very quickly. Then you don't want to play with it. Or it is hot out or bugs and you don't want to go out and spend time with it because of those inconveniences. Then it becomes so attention seeking that it literally jumps at every opportunity to spend any time with you whatsoever, so you avoid it because it jumps on you and is unmanageable. It just gets worse and worse until you don't know why you have the dog in the first place.
The only way (imho) to really enjoy the rewards of pet ownership is to take them into your home inside with you. You and your daughter will become your pets family! And it will become part of yours.
I had a cat when I was single, and it was great. Until it began spraying (he was fixed for years before he started this) because of neighboring cats sniffing around my apartment door. Then it was a big mess with the carpet. Otherwise, I really enjoying having him as a companion for several years.
My roommate had a dog at one point, and I HATED it... she did not take the time to properly train the dog to stay out of the trash, stay off the furniture, etc. And the house began smelling b/c she didn't have her spayed either.
So I would never have guessed that one day when my husband and I decided our kids needed pets, that we would opt for an inside dog. And a big breed at that. I ADORE her. We did extensive research into the breed itself and how it would fit into our family lifestyle. Also several different training methods (which is crucial to understand right from the start... otherwise you have to un-teach or re-teach things and that is much harder). We learned about crate training, and how wonderful it can be when done properly. We have had our canine family member for 6 years now (next week actually will be exactly 6 yrs) and I wouldn't trade her for anything.
You didn't say how old your daughter is, but unless she is at least 10 or 12, I would not have any expectations of her having any real meaningful responsibility with your pet. Especially a dog. It will all be on you. A cat, not nearly the same responsibility... even a 6 yr old might be able to scoop the litter, drop it into the toilet and flush.
You could also consider a gerbil or guinea pig. I have heard that they make wonderful pets for children as well. When properly socialized they like to cuddle too.
Oh, wanted to add about scooping the poo... often times dogs' poo is (WARNING: TMI ! lol) big and mushy and very smelly because of a poor quality diet. We started out giving Iams (b/c I thought it was a good quality brand) and our dog went about 3 x a day. My SIL used Purina One, so we tried it... and it made a vast improvement. Now she only goes about once, maybe twice a day, and it is smaller and more compact, (less fillers in the food makes for less waste) and doesn't have the extreme odors, either.
Like people, diet has a major impact on "output". :)
FWIW, our dog is a German Shepherd. If you are leaning toward a dog, if you can "deal" with a little hair under the furniture, they are the BEST. Ours LOVES our kids and even babies she has never met before. She is SOOO gentle with them. As a puppy, she was quite hyper. But she did outgrow that and mostly lays around sleeping unless someone is running through the house or goes outside. She never chewed much of anything, either.. which there ARE breeds that are notorious for damaging by chewing. Even when she was teething (and ALL dogs chew then) she only chewed up a few inexpensive plastic toys the kids left lying around... no furniture, no baseboards, no walls, no curtains, no shoes, etc etc. She is SO smart and tidy. She doesn't even make a mess when she eats... she licks up any crumbs that drop on the floor around her dish. She sleeps on the floor alongside my side of the bed (her choosing) and it is such a comfort to have her there. Especially when hubby works late or has night shifts.