Hello,
I too have seen this exact problem with ESXi 6.5, with the latest patch (released March 14th 2017) installed. However, I only have it when I hit "enter" after typing in the password. I do not get the error when I click "Log In".
I can completely re-create this error on 2 different ESXi 6.5 systems.
Hit enter after password: "unhandled exception" error.
Click Login: Everything works perfectly.
This issue does exist on Chrome (latest version as of March 20th), but does not exist on Microsoft Edge, the other browser I have installed. So at this moment I am deducting it to be one of two problems: 1) There is a problem with Chrome and ESXi web client that needs to be fixed in Chrome, or 2) There is a problem in the ESXi web client that needs to be fixed.
Since this problem does not exist on any other HTML5 websites in Chrome, I am inclined to think there is a coding error in the ESXi web client that has not yet been found.
I am not impacted as clicking "Log In" works.... but oh man, I do wish VMWare did not abandon the "thick client" idea.....