Even though there is no option to hide the attachments section by default, you are able to hide it using Custom CSS. To do so, navigate to Configuration > Chat > Chat buttons > Edit your chat button and open Pre-chat form > Design. Now scroll down to the "Custom CSS" section and add the following CSS code:
.MessageFormFilesWrapper, .MessageArea .DragAndDropFileUploadView {
display: none !important;
}
Do not forget to APPLY the CSS code and SAVE the design.