With ONLY javascript and angular, in a normal web browser you'd expect a consumer to have installed this wouldn't be possible.
All of the mainstream browsers offer dev tools where you can inspect the variables in use in the browser. So a breakpoint in the right spot will grant anyone with the web page open access to the variable. Also once the data is in the clipboard you've lost control of the data anyway, and it's exposed for other applications to copy from the clipboard. It sounds like you may need some sort of locally installed thin client (desktop app), to securely allow the user to interact with this data.
Read here about memory management in the javascript virtual machine to get an idea about when and how memory is cleaned up https://developer.mozilla.org/en-US/docs/Web/JavaScript/Memory_Management
Maybe someone out there knows a way to do this with javascript, but I don't believe it to be possible, but it's hard to know what you don't know.
The .NET SecureString does things for you that just wouldn't work in a javascript VM that is client side (not node.js on the server, that's a different scenario). If you did have such a feature in a client side javascript VM, you would need a way to encrypt the string data, which means somewhere on the client or in your web page you need an encryption key and initialization vector that have to be stored somewhere, but more importantly they need to be protected.
Given all the data is on the client machine and isn't leaving the client machine, what threats are you trying to protect this data from? Is the user transferring the data from the clipboard to a remote computer? What is supposed to happen with the data after it's in the clipboard?