Allow rented properties to allow pets

I am young woman and suffer with depression and just been told I may have bipolar disorder. I get low and I can't deal with being alone I'm being told by many people that way to live with it is having something to focus on like a small dog or a pet that will need to be look after.
A dog would be perfect as can get out the house and will have to walk it.
One problem is im unfortunately not able to buy a house and never really would be able to. The onley option is living in rented acomadation but your not allowed pets in rented acomadation.
I think this is wrong in so many ways having pets come with some great rewards it lowers depression and keeps you healthy and is a good comfort and destress to have pets to come home to.
To many shelters are over full mainley due to people suddenly having to move and go into a property that don't allow pets which to some people pets are people famaily some people can't have kids and all they got is there pet and just because of what ever circumstances they have to give them up because rented properties won't allow you to have pets. This is wrong you would not say to someone with a child or children that they have to put their kids into care due to needed somewhere to live don't allow children.
I think if the law change so that you can have pets as long as you pay for any damage cause by your pet and take full responsibility instead being told no I think it would lower the amount of dogs and cats ect in shelters or being dump and lower the amount of people who suffer with mental health issue such as depression and sucidal thoughts I feel it be a all round improvement if this was allowed to happen I don't understand how people in council or own there own homes can have freedom to have pets but people who rent can't. I understand of the fear that properties can be damage but if you rent a property hoping to stay long term as long as you pay rent and clean up then I don't think there should be a issue