Is it possible to display another column in Entries type field? I would love to be able to show another column inside Entries type field. Specifically, Image field. Basically,the same functionality as on the Entries list. Is there a plugin or another way of implementing this.
Nope, not out of the box – but being able to display fields in addition to the entry title for selected entries being displayed in an Entries field, in an easy-to-use interface like the Customizable Element Index, sounds like a killer feature request.

For now, your best (albeit not super easy, I suspect) bet would be to create a custom plugin that adds a custom FieldType extending the EntriesFieldType class.

By overriding the BaseElementFieldType's $inputTemplate property (i.e. by adding an $inputTemplate property to your custom FieldType's class, pointing to a different template) you might be able to display entries in a different manner compared to the vanilla Entries field.