So our dog is 16 or 17 years old, I cant remember. Hes a shit zu yorkie mix, so hes very small, and very old for his size. Hes also completely blind, completely deaf, has multiple bladder infections which he needs daily medications for, and has weak back legs.
My parents have recently made a lil caged spot for him to wander around and pee in since we cant watch over him 24/7, he pees and poos on the floor all the time due to his infection, cant hear us calling to him, bumps into walls and furniture constantly, not to mention falls down the stairs when he is let out for a bit, and whenever hes in his little "zone" he cries and yowls like hes in pain every single minute.

My sister is the one who really owns him, and I want to tell her that I think it's time he gets put to sleep, since hes clearly not happy, but she loves the poor old boy to death, and I feel like all that would do is piss her off. Even though I feel as if she doesnt see that her favourite puppy is gone and all that's left is a miserable little dog, as if she's a kid who refuses to get rid of her oldest toy and plays with it over and over until it eventually breaks.
She'd probably think "you dont understand, you dont love him like I do, you just want him dead so his crying doesnt wake you up at 6am or so you don't have to clean his pee puddles anymore"

Idk how I should say it to her. Idk what to do here, if you guys have any ideas then please tell me, even if there is a way to keep him around longer than that's ok

Having an animal put down is a tough call, and it’s hard to tell someone else when to make that call. From what you’re describing- if it were my dog I would ask a few simple questions. 1. If he didn’t have the infection, would he be howling in pain? Would he be peeing all over?
2. What are his odds of recovery from the infection?
3. What is the time table and prognosis of when he should recover or at what point we might know for sure he won’t?
4. If the infection gets worse, how long and how bad will things go as it progresses?
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If he’s got fair odds of recovering, and maybe not being a puppy or even having much mobility but being able to live and not be in constant pain- then hey- everyone gets old and that’s fine if that’s a persons choice. Hell- some people would rather be euthanized once they get old enough and some want to stay holding on to The end. But if the poor thing will be in escalating misery for its remaining time on earth- it’s likely time.

So how do you bring it up? I don’t know your sister, or the relationship y’all have. So take my opinion with a grain of salt. But I figure you’re better off not coming out of the blue, and not using a conversation that is emotionally negative to start as your opening. You’re also best suited to not bring it up right after school or work, or when she’s in a bad mood. Feel that part out with softballs and go deeper as you get positive responses. I wouldn’t tell her. I’d ask her. I’d ask from a place of concern and let her know that I’m worried and interested in the dogs health and what is going on. From there it’s pretty easy to find an in to ask her if she’s thought of other options if things continue as is or get worse, and if she’s set a time frame. If she loves this dog, then she wants what is best for the dog and not for her. And I’d say it if needed- that we need to think about the dog and what this is like for it. It’s health is hard on everyone to go through, seeing it and I’m...

... sure living it too. At some point it comes time to say goodbye, and that’s hard. But if saying goodbye a little sooner on your terms allows the dog to be out of its pain, and allows you to not have your lasting memories be of a sad and miserable animal barely alive and suffering, it may be best. Of course- some people just don’t believe in it. They feel like they wouldn’t do it to a person, and some feel like they would. Others think it’s ok for animals and not people. If she doesn’t believe in to that is her way. She’s got to carry that. But if you talk to her, or depending on age and family structure- your parents, if talking to her goes poorly, you’ll have done all you could. In the end if she does tell you that you don’t care about the dog as much as she does- I can’t argue it because I don’t know, and you likely shouldn’t argue it. I’d tell her the truth. It doesn’t matter who cares about the dog more than who because it’s about more than one persons feeling, about more than..

.. her feelings. Regardless of who loves the dog the most you all care about it or you wouldn’t be havin the talk. You care about her too or you wouldn’t be facing her, and it’s hard on you too watching it suffer and hard for you to have brought it up with her and I’d tell her at that point all of that. That it stopped being about just her feelings the moment she took that dog in, that she is responsible for it. It loves her and trusts her and that every moment it is in pain she is in pain, you are in pain and that dragging that pain out of there isn’t a hope it will get better is cruel to everyone. Tell her you love her and you’re sorry that this had to happen, that there’s little worse than knowing that someone or something you love is hurting and you can’t do anything about it. But she can do something about it. She can make her friend no longer hurt, and that puppy will live on the same place it would a year or whatever from now if she lets the dog suffer until the end- in memories

I agree with everything @_guest said. Sometimes truly loving something is letting it go.

Pretty much. Owning a pet is making a promise to that animal to ensure you'll do everything in your power to make certain it has a good quality of life. If that means giving it medicine that tastes awful, you do that. If it means getting the earwax out of its ears, or giving it baths, or clipping it's toenails, you do that. And if it means realizing the animal is suffering and needs to be let go, you have to make that call.
It's not easy. It's NEVER easy. But the animal is relying on you. Forcing it to stay alive until disease waste it away is not kindness.

What’s the veterinarian’s opinion?

Not sure, I wasn't there when we found out about his infection

UPDATE
So good and bad news, we scheduled it. We're gonna say goodbye on Friday. This is good bc it means he'll no longer be suffering but bad bc... well, we'll all miss him

I’m sorry. It’s tough. He’s lived though- because real love means doing things- even the last thing we’d ever want to do. The hardest thing- because it’s what is best for the one we love and not for us. It is putting another ahead of yourself, even when to do so is almost as painful a thought as death itself. He was a good boy and I never had to meet him to know it, but will never know just how good he is. That’s going to live in your memories and your stories- and in this way he may outlive us all. For the love of a dog is like no other. It is the truest love of a creature that will love another more than it could ever love itself. Through the years and in this act you honor that love, repay that love. For what it is worth- my best to you and yours in this hard time. You’ll send him off right and he will be loved to the very end. That is a good life. A better life than many will ever know and it was all possible because of you. We could all hope to be so lucky.

Are you doing okay? I know this is the best decision, but that doesn't necessarily mean it's not hard

I'm alright, I'm more worried about my sister than me

Something we would always do with dogs was save a lock of fur or an impression of a paw print. For appreciating and remembering. Better to go out painlessly, I suppose. He’ll know he was loved up until the end. Good luck.

Hopefully she knows and understands it's the best thing for the dog. Even still. Anytime my family had to put down an animal as bad as I'd feel I'd always be worried about my mom because she'd take it so very hard. There's not a whole lot that can be done besides just being there, sadly. As silvermyth said, you can also get an impression paw print. They can help somewhat with the grieving process I found, but that may be a personal thing