In order to fill the page, the page needs to fill the window:
styles.css
body, html{
  height: 100%;
  margin: 0;
}
The app component needs to fill the body:
app.component.css
:host{
  box-sizing: border-box;
  display: block;
  height: 100%;
}
By default the wrappers of the content within the tabs are not filling the tab's height so we will fix this:
styles.css
.mat-tab-body-wrapper{
  flex-grow: 1;
}
We also need to have the mat-tab-group fill the height, its parent needs to fill the height too, so we will give an id to both the wrapper and the tab-group.
app.component.html
<div id="wrapper"> <!-- give it an id -->
    <div fxLayout="row" fxLayoutGap="16px" fxFill>
        <div fxFlex="20">
            <p>Fun Sidebar</p>
        </div>
        <div fxFlex="80" fxFill>
            <mat-tab-group id="tab-group"> <!-- give it an id -->
                <mat-tab label="Summary">
                    <div fxFlex style="padding: 16px;">
                        <div class="mat-display-2">Hello</div>
                        <p>Lorem ipsulem</p>
                    </div>
                </mat-tab>
                <mat-tab label="Tall content">
                    <app-tall-component></app-tall-component>
                </mat-tab>
            </mat-tab-group>
        </div>
    </div>
</div>
app.component.css
#wrapper{
  padding: 16px; 
  min-height: 100%; 
  height: 100%; 
  box-sizing: border-box;/*new*/
}
#tab-group{
  height: 100%;
}
#tab-group mat-tab-body {
  flex-grow: 1;
}
https://stackblitz.com/edit/angular-awamwn?file=src%2Fapp%2Fapp.component.html