I think part of why so many people here dislike it is because we have a lot of introverts on here compared to the general population. Plus I’m reminded of the Onion’s take on modern society’s preference for impersonal shopping experiences. 
What benefit would that provide aside from delaying what you likely need to face? fluiddruid’s advice seems a very sensible and reasonable approach. I appreciate that you may be dreading what could be an uncomfortable encounter, but delaying will not make it easier, nor make her feel any better about it when you put the necessary boundaries in place.
I’m in commissioned sales, and I’m hardly an introvert so your lazy characterization of people’s probable objection to loiterers is that dopers just don’t like to socialize is a bit off the mark.
Here’s the scoop. If I go into a store and I see the same non-employee person just hanging out time after time that’s odd and unprofessional in a modern store. Some people may think it’s charming I do not. I worked as a Radio Shack manager for 5 years right after college in the early 80’s. There were always young kids that wanted to hang out for hours, and play with the (then new) computers. Some managers thought this was OK, but in the end it always turned into a mess because the kids would monopolize the display computers, and in some cases start giving customers advice to shop elsewhere for the best deals on diskettes etc. It was like a little clubhouse in some stores. There were also theft issues in some cases.
You’re either working for a store or you’re not. If you taking up time and space just hanging out and the manager is letting that go they are creating an unprofessional environment. Some people might like that, but I don’t want to shop there.
I think fluiddruid’s advice was perfect. The only thing I’d say is that whatever you do, you need to make sure your co-owner is following through as well. You need a united front here.
To me she sounds a little mentally unbalanced and would be afraid that she is really seeing you as a close friend, and that next she might start showing up at your house. I would follow some of the advice here and put a stop to this now, before it gets worse.
Does your local animal shelter take volunteers?
If so, you could tell her you think she is suitable.
It really depends on what the shop is though- what you say is pretty well on the mark for an electronics store, but I’d be worried if I went into a comic book shop and it didn’t have a few regulars hanging around.
One of the pet shops near where I used to live was inhabited by a volunteer for ages- a lady I knew, who was a retired Veterinary surgeon and had written books on reptile keeping. She was a lot more knowledgeable than the staff, and only didn’t work there paid because she didn’t need the money, and liked having the freedom of going ‘screw this, it’s quiet, and I’m going home’, as well as being able to pick which jobs she wanted to do with impunity.
That sort of situation happens a lot when dealing with animals; though it should only happen when everyone is happy with the arrangement, which clearly isn’t the case here, but it’s understandable that a socially clueless person could think it is. She’s ‘helping’.
I think she’s not doing anything wrong exactly, just in this case it’s very unwelcome. Getting her to go away (or at least cut down visits to a reasonable length, she does need bird food after all) with the minimum of nastiness should be the goal. It’s tough to do though.
Can you send her to the local animal shelter/ animal welfare group instead? They probably need somebody to look after the birds in the shelter.
I agree that fluiddruid’s advice is spot on.
However, if you’d like to try something less confrontational, as a first line of defense, and which may or may not work given her cluelessness:
The minute she walks in the store, and every time she does, walk straight up to her and say, “Hello, Mrs. M. Is there something I can help you find today to get you on your way?”
Key point here, use a formal form of address (Mrs. or Ms. whatever, or ma’am if nothing else), not her first name. You’re not being her friend, you’re being a professional shopkeeper. Also, make it clear that you think that this is going to be a short shopping trip. Establish that and she may feel a certain amount of pressure to meet your expectations.
If she says yes, help her find it. Don’t let her change the subject or meander onto a tangential topic, bring the conversation back to the item she’s buying, then bring it to the counter, check her out, and show her to the door.
If she says no, say “Thanks for stopping by, then. Have a great day.” And show her to the door.
In other words, don’t give her any opportunity to cross a boundary. Get her taken care of and out the door. If she tries to stay, tell her “I’m sure you have other things to take care of today, so I won’t keep you.”
As I said, this may not work depending on how stubborn and clueless she is. In which case, the direct approach is your best bet.
(And I personally wouldn’t send her to bother someone else. I doubt they’d appreciate her loitering either, and it’s really not your job to fix her life for her.)
Thank you everyone for your answers. It’s hard to confront a customer sometimes, but this is definitely a case where it needs to happen. She’s not really outwardly disruptive, but her presence is too much!
I’m still astounded that she spends hours of her day in your store, spent 5 hours in your store in one day, goes behind the counter, and you didn’t boot her out and you wonder if you should do something about it. Maybe it all happened so gradually you didn’t realize how weird it is.
Ask her.
Look, I realize it’s weird. I came in here to ask how I should handle it, not to determine if it was strange. I have regulars in my store, but she goes beyond the normal definition. It started normal, and I do think it gradually slid into this weird relationship we have now. I admitted I’m a wimp, but that’s not entirely true. It’s really hard to confront someone in person when they obviously mean well. I’m not defending her here, she’s crazy as a bedbug, but it’s tough to tell her to leave the store when she’s so well-meaning. I don’t enjoy hurting people who are so apparently lonely.
You mentioned in one of your posts that you wonder about her living conditions. What do you know about that? Is it possible that her home has no heat or water, thus she is using your store as a place to stay warm and the use of a bathroom? If so, perhaps your county or local social services could get involved?
You remind me of a few very nice women at my gym who feel sorry for lonely old men and will let them interrupt their workouts to the point that they get nothing accomplished! I think that they are just too nice, and you are too.
Good luck! You can do it. 
Actually, I see people like this at the big box places like Petco that come in with their dogs for visits on a regular basis and stick around for a long time. Think I’ve seen some crazy cat ladies and some Bird Ladies, too. And for their store that is a retail feature and its sort of a place for animal lovers to congregrate with their animals… so in that case at least it is a retail plus, and doesn’t really jibe with others who seem to see it as a minus in the retail pet game. Just sayin’, if a small pet shop wants to compete in today’s big box retail game they better at least tolerate the Crazy dog, bird, and cat people and even encourage them like Petco does, that’s the core of their business… and if you don’t want her, maybe suggest a place like that to her. I’m sure they would tolerate her better and even welcome her (I mean you did say she usually buys something, right?)
Did you read the bit where this woman goes behind the counter, and uses the restroom without permission, and things like that? She’s no longer a customer, she’s a liability. A customer has absolutely NO reason to go behind the counter in everyday retail activity. About the only reason I can think of when it would be appropriate for a customer to go behind the counter is if one of the workers was having a heart attack or something.
For what it’s worth, I know some musicians/store owners who will give a ‘customer’ a very rude get-the-f***-out-or-buy-something response if they hang around and play instruments for a couple hours…and that’s just for one day. Bigger chain stores seem to be more tolerant of lingerers though.
Ask if she made any New Year’s Resolutions - and mention that your is to spend your time in the shop when there are no customers catching up on your paperwork/reading/plans etc., and when there are customers you will be the one to serve them. Then when she is dawdling you can remind her that you have work to do and need the empty shop time. This is in addition to the previous advice to tell her she can no longer use the restroom, open cages, and come behind the counter.
Well, “behind the counter” at the big box stores is just another queue in the line of “retail checkout lines” and they have public restrooms for people as well as “accident” disposal and clean up stations for visiting pets. jus’ sayin’.
The big box just seems, I don’t know… alot less judgemental.