<button mat-button [matMenuTriggerFor]="menu">Menu</button>
    <mat-menu #menu="matMenu" backdropClass="custom__menu">
      <button mat-menu-item>Item 1</button>
      <button mat-menu-item>Item 2</button>
    </mat-menu>

    .custom__menu + * .cdk-overlay-pane > div.mat-menu-panel > div.mat-menu-content {
     ... // Add whatever styling you want
    }