Custom sharing button doesn’t work

Same.

Also broken on this tutorial on the same subject:

https://wphow.org/how-to-add-a-custom-sharing-service-to-jetpack/#themify_builder_content-3302

Thanks for the report.

This will be fixed in the next Jetpack release, that will be available in a couple of weeks. Until then, you can apply this patch if you feel comfortable edit PHP files on your site:
https://github.com/Automattic/jetpack/pull/6355