It seems to be an open source reusable component that you can include in your apps via code (look at this sample). but the component is not implementing any of the interfaces by default to enable it to be used in the app builder or lightning actions or community builder etc..
you can edit the component code in your org and make it implement the interface flexipage:availableForAllPageTypes for it to appear in the App builder..
you can go thru this article for details on how to enable components for App builder.
Installing beta package releases into production orgs is not possible except in very special cases (the "production" orgs created via the partner portal), so the answer to your first question is almost always no.
For your second question, you can bypass the appexchange if you want, and distribute your app directly via it's install link, certainly. If you don't go through security review some benefits of being a managed app are witheld however. For example, if you go through security review tabs in your app don't count against the customer's tab limit for their org, but if you skip the review they do count.
You can still use the license management app with a non-reviewed application by creating a private appexchange listing and using that to manage the license settings, however going through security review is recommended of course.
Push upgrades are only available to apps that have undergone security review, and once that's done have to be enabled by salesforce support for your dev org.
I created a new version and I pushed to the customer again, and seems to solve the issue, somehow it shows the component in his org. Thank you for all the answers.
Updated on: 2019-09-27
Another way to solve this issue is to compile all the Apex classes.
Best Answer
Looking at the screenshot and searching for the Lightning Tree component in appexchange led me to this github source for the component
It seems to be an open source reusable component that you can include in your apps via code (look at this sample). but the component is not implementing any of the interfaces by default to enable it to be used in the app builder or lightning actions or community builder etc..
you can edit the component code in your org and make it implement the interface flexipage:availableForAllPageTypes for it to appear in the App builder..
you can go thru this article for details on how to enable components for App builder.