This story has a lot to do with a theory. The Andy's mom is Emily, Jessie's previous owner. Its a complicated theory, so just look it up if you've never heard of it. So this story is that that theory is real as well as the theory that Andy's DAD gave him Woody. As for why Woody doesn't remember his old owner? I don't know. Just go with the theory. Maybe you'll end up believing it.
They were little once. Everyone was. Emily was little once. A little girl who liked playing with cowgirl dolls. She loved her Jessie doll. A lot. She liked playing cowgirl and running around on her little horse on a stick. She spent her days pretending to be who she wasn't and making friends who were just as keen to pretending. She met a little boy who liked pretending. Pretending was all he had. Well that and a woody doll. The little boy had grown up in a house full of turmoil. His parents were always fighting and eventually went through a divorce. He held on to his pretending and his toy and his new friend and looked toward the cowboy stories they made up together for the comfort he didn't find at home. They played pretend with each other every day as they grew older until they grew too old, or at least she did. She became interested in teenage girl stuff that confused him. He still spent time with her, she was his best friend, but when they weren't together he didn't give up on the make believe stories he held on to as a child. He began writing them down, turning them into full fledged books, using the Woody doll he still held on to as inspiration. He didn't tell her about it though. She would probably laugh. She was too old too mature for pretending anymore. She talked on the phone and sang along to the music on the radio. She didn't know how cute she was when she did it either. He didn't have the guts to tell her. So he just hid in his room and wrote and went to his dad's every other weekend. Even though he was getting too old he still brought Woody.
One day he found the guts to tell her how cute she was. She blushed and suddenly everything changed. Suddenly they were dating and everything had changed. He was too busy to write most of the time. For months Woody sat on the shelf only earning a sympathetic glance from his owner every now and then. One day when Emily came over she saw the old toy. She laughed.
"You still have this?"
"Yeah. You and Woody got me through some of the hardest times of my life." He said, picking up Woody, straightening his hat and dusting him off.
"You're so weird you know. Its just a toy." No its not, something inside of him said believing Woody to be so much more.
"Yeah whatever." He said. He couldn't argue with his girlfriend.
They ended up getting married. Settling down. Having a kid. Having another. But she didn't understand didn't get his writing or why she wasn't his inspiration. It got too stressful for him, besides he hadn't had such good marriage role models. He ended up getting a divorce, deserting his own son just like his father had deserted him. But he didn't make Andy go it alone. He left a gift. Sheriff Woody, a toy who existed to get people through hard times.
He and Emily had been little once, playing pretend, making all the bad things go away just by pretending to be someone else. Emily had stopped pretending, grew up and gave up on their childhood. He never had, and hoped Andy never would either. He wanted Woody to help his son like it had helped him. Maybe help little kids after Andy too.
"Don't grow up." He whispered to his son as he tucked Woody into Andy's arms as Andy slept.
"Always be little." He begged as he left his son's room for the last time.
Better than i thought it would be.