First of all, I will define what ownership means from my perspective as a software engineer.
Ownership, the actions that the software engineers do because they know that their work is crucial to the success of the project, team, and the company they are working at; It's not only about actions, feelings and passion are important as well here.

Ownership will force you to learn more

If you have ownership then you are committed to the best quality you can deliver; When you commit to quality you will start reading more about your deliverables, about the technology stack you are using, what makes it better, etc.

For example
if you are a frontend engineer that your task is to create a contact form.
If you just want to deliver it you can do that in no time, but if you are committed to the best quality I am sure you will start searching for other contact forms, maybe you will stumble upon some weird terms such as a11y, A/B testing, performance budget, or analytics.

So your "Ownership" leads you to learn more to deliver the best quality.

Ownership will make you a better team player

Teamwork is a quality attribute of any software engineer, to the point that we stopped writing it down in our CVs because you have to have it as much as having good problem-solving skills.

Ownership will force you to communicate with your teammates and go far and beyond to help them get the work done because you are a crucial part of their success and you know it.

Owners will not point fingers, they will jump in to help because they care about how to make it work rather than blaming someone. And as humans, we love to be helped so we have to help to earn that.

Ownership will enhance your debugging and engineering skills !!

Debugging skills is essential for any software engineer. because we build both software and bugs, the beauty of it is that we KNOW it.

Imagine you are at home watching Netflix at 2:00 AM, and boom slack notification in your production issues channel; without thinking you will go and check it out and you replied I am taking a look.
After your initial investigation, you found that the bug is data returned in a wrong format from BE.
You put a quick fix in the FE to unblock the customers - because you care - and then you went back to trace WHY it's wrong as a Frontend engineer you reached the ORM class and found the root cause.
You went through 3 or 4 microservices, and found the root cause!! You are CHOSEN!!
And simply sent a slack message to the engineer working on the ORM, and said hey I think the incident is related to those lines, next day at work she will buy you coffee and respect what you did

You debug because you are committed to the success of everyone around you, and it's a brain training the more you do the better you will be.

Ownership will make you take more responsibilities

The more ownership you have the more you will know, the more you are respected among your teammates, and in your company; That will make you a go-to person in some cases even if you don't know anything about that topic you will start being curious about it, check the "LEARN MORE"

More responsibilities mean a bigger role, more challenging tasks, and maybe a raise :P.

Summary

For whatever reason that doesn't make you feel like an owner, I have to jump in here and tell you to do it even if it for yourself not for anyone else, you will grow no matter what.

There is a myth that says, If sharks stopped moving they will die.
Keep moving for your own sake you will grow and make people around you grow as well, you will make

Have the faith that if you do good you will be rewarded, in your current position or somewhere else.
Ownership will make you a better software engineer because you will:

1. Learn more.
2. Become a better team player.
3. It will enhance your debugging and engineering skills.
4. It will open the path for more responsibilities.