Is it possible for a progressive web app to embed web manifest into html page?

There's nothing wrong with the Data URI in your question. It works.

However, it contains an invalid JSON.

The JSON is invalid because it begins with:

<%@ Page ContentType="application/json" %>

If, instead, you use:

<link rel="manifest" href="data:application/manifest+json;base64,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">

(which is the same as what you have, but with that first JSON-invalid line removed)

it will work.

Plaintext Alternative:

Alternatively, you don't even need to base-64 encode the data:

<link rel="manifest" href='data:application/manifest+json,{"name":"SharePoint Title","short_name":"SP Title","description":"The one and only SharePoint Site Title program!","start_url":"index.html","icons":[{"src":"android-chrome-144x144.png","sizes":"144x144","type":"image/png"},{"src":"android-chrome-192x192.png","sizes":"192x192","type":"image/png"},{"src":"android-chrome-256x256.png","sizes":"256x256","type":"image/png"}],"background":"rgb(255,0,0)","theme_color":"rgb(255,255,255)","display":"standalone"}' />

No

The manifest file must be a separate file. It also needs to have the proper Content-Type header - application/json; charset=utf-8

It can be gzipped.

You want it to be separate because the browser will only download it once (per version of course) and use it to direct the way your PWA is presented. This is very similar to the way Live Tile support in Windows has worked for years now. However with Live Tiles you could inline properties as META tag values.

You are much better off having a separate file because it would be repeated overhead that can be avoided each time a page is requested. Now the end user only downloads the content once, not every time.