A blog post by Siderite (Styling Angular Material tooltips) provided an answer that worked for me. I am paraphrasing from the most-relevant portion of his post and I am using the matTooltipClass="tooltip" scenario described in the Question above:
[The .tooltip class definition] should either be in the global CSS file (so it applies to everything) or your component should declare encapsulation ViewEncapsulation.None. [If the .tooltip class definition is in the global CSS file], then ensure the declaration of the class is specific enough: try this: mat-tooltip-component .mat-tooltip.tooltip {...}.
In my case, I had defined the .tooltip class in the global styles.scss file, and it wasn't working until I followed Siderite's suggestion and defined it like this:
mat-tooltip-component .mat-tooltip.tooltip { color: blue !important; }
This approach is avoids using ::ng-deep as suggested in the accepted Answer. Angular documentation states that approach is deprecated. I did find I needed to use !important, which some believe is bad style.