My English isn't that great so please bear it with me. My interpretation:
I'm pretty sure that you know that in Japan, 20th birthday is an important day as it's the first day of adulthood.
However, during the year before that day, adults tend to pressure -- via questions, advice, suggestions and warnings -- on 19-year-olds, to think and plan their future, and what they will do on their 20th birthday. Almost all adults warn that you'll ruin your life if you don't plan your future with care. You might end up living the worst scenario of your life if you're not careful.
That kind of pressure makes 19-year-olds competitive, anxious, excited, tense, rebellious, unrealistically ambitious, cautious, restless, hopeless, indifferent or whatnot. However, almost all share one common feeling: it feels as if they're heading into something that might trap them for life. Kind of like being a mouse in a maze with adults as walls that guide the mouse to the only door available. Some resent this, some don't and some need it.
Regardless, almost all look forward to doing something to make their 20th birthday memorable. Some take a weekend vacation with friends, who also celebrate their 20th. Some attend an official ceremony on Coming of Age Day during early January. The lucky ones have their parents paying a sport car, expensive vacation, apartment or an opportunity to attend a university overseas to mark the day. Some keep it simple by going out for drinks with good friends or have a dinner party at home.
Hence, the old man reacts with surprise when Birthday Girl tells him it's her birthday. It's her 20th birthday and yet she's treating it as just an ordinary day? Unusual.
There are clues that Birthday Girl is trying to escape from her fate as a trapped mouse. She lives far from her parents. Broke up with her childhood sweetheart. Doesn't inform her manager it's her 20th birthday. Doesn't tell her coworkers, either. And of course, there's her own reaction after the old man grants her wish.
I believe her wish is that she wants to take each day as it is and leave the rest up to fate.
Most would consider that irresponsible. Or predict that it'd put her on a road to the worst nightmare possible. Such as ending up poor and unemployed with a sick baby in the worst part of a town.
I think for her, the old man granting her wish is akin to giving her permission to trust her instincts, such as allowing her desire to live each day rather than plan for tomorrow. The clue to this is when she leaves her job the day after New Year's Day. This suggests she will attend Coming of Age ceremony with a clear idea of what she wants for her future.
As she says to the narrator, her life turns out okay. She's happily married with two kids, a good job and a car with a couple of dents in its bumper. Dark shadows haven't visited her as a result of not planning her future. Sounds like her wish came true.
When the narrator couldn't think of a wish, it suggests he's also successfully escaped from his fate as a trapped mouse. Hence, her last line.