I came across below question where the solution for IE 8 is provided.
Call .Net assembly from OBJECT tag in IE8
Is there any solution or workaround for the Object tag code for Microsoft Edge browser. The .net application is built in .Net 2.0 Framework. it is working fine now till IE7, but as Internet Explorer browser might be removed it should get rendered on Microsoft Edge.
ExamineFile.dll is a dll of class library containing a Windows Form and related source code.
<object id='ExamineFile' name='ExamineFile' classid='ExamineFile.dll#ExamineFile.FileInfo' ></object>
UPDATE
I updated HTML to HTML 5 and modified tag as below
<object id='ExamineFile' name='ExamineFile' data='ExamineFile.dll#ExamineFile.FileInfo' type='application/x-msdownload' ></object>
I am getting below result now.
Based on my research, it was possible to do with the older versions of the IE browser.
It is not possible to call .Net assembly from the Object tag in the MS Edge browser.
I did not get any direct workaround for the said requirement.
If your goal is to make a connection between the web application and .Net assembly then you can try to check WebView2 and Getting started with WebView2. It may help you with your issue.