Please see 'Embed a Visio drawing in a classic SharePoint page with key properties' section in this article https://blogs.msdn.microsoft.com/visio/2017/09/25/migrate-from-visio-web-access-to-visio-online for a code solution to achieve the above requirement.
Visio Online provides the capability of embedding in a SharePoint page, through Document Web Part. We are planning to add basic customization properties to the webpart in future.
Please see Visio Online FAQs for more information regarding above.
Thanks for your feedback.
Visio Online supports relative hyperlinks and sub-addresses as well.