Thanks for the testing page, much appreciated.
When you work in the Visual tab on a page or post, WordPress is turning line breaks into paragraphs automatically. Text widget’s often get used to insert HTML, as a result they don’t insert paragraphs automatically and provide a checkbox at the bottom of the widget, leaving the setting as a user option. If the paragraphs checkbox isn’t checked you’d need to insert them manually as follows:
<p>
Paragraph text.
</p>
Or if you needed a line break within a paragraph, you’d say:
<p>Paragraph text.
<br />
And now we're on the next line thanks to the break tag above.
</p>
In conclusion, the behaviour we’re seeing here is what I’d expect to see from a Text widget with and without the paragraph setting checked.
If you’d like the ease of editing text in widget areas using a Visual Editor (the same one you have on pages/posts) then I’d recommend trying Black Studio TinyMCE: https://www.ads-software.com/plugins/black-studio-tinymce-widget/. Once installed the plugin will add a widget called Visual Editor. To confirm, the widget this plugin provides is not the same name as the plugin itself. Definitely one of my favourite WordPress widgets.
Hopefully that clears things up a little. Please let me know if you have any questions about this or any other topic.
