Can I change the default mat-paginator size depending on screensize?

You can use onresize event : (window:resize)="onResize($event)"

<div class="mat-elevation-z8" (window:resize)="onResize($event)">
  <table mat-table [dataSource]="dataSource" matSort>

    <!-- ID Column -->
    <ng-container matColumnDef="id">
      <th mat-header-cell *matHeaderCellDef mat-sort-header> ID </th>
      <td mat-cell *matCellDef="let row"> {{row.id}} </td>
    </ng-container>

    <!-- Progress Column -->
    <ng-container matColumnDef="progress">
      <th mat-header-cell *matHeaderCellDef mat-sort-header> Progress </th>
      <td mat-cell *matCellDef="let row"> {{row.progress}}% </td>
    </ng-container>

    <!-- Name Column -->
    <ng-container matColumnDef="name">
      <th mat-header-cell *matHeaderCellDef mat-sort-header> Name </th>
      <td mat-cell *matCellDef="let row"> {{row.name}} </td>
    </ng-container>

    <!-- Color Column -->
    <ng-container matColumnDef="color">
      <th mat-header-cell *matHeaderCellDef mat-sort-header> Color </th>
      <td mat-cell *matCellDef="let row" [style.color]="row.color"> {{row.color}} </td>
    </ng-container>

    <tr mat-header-row *matHeaderRowDef="displayedColumns"></tr>
    <tr mat-row *matRowDef="let row; columns: displayedColumns;">
    </tr>
  </table>

  <mat-paginator [pageSize]="pageSize" [pageSizeOptions]="[5, 10, 15, 20, 25, dataSource.data.length]"></mat-paginator>
</div>

//your component

  onResize(event) {

    console.log(event.target.innerWidth);
    if (event.target.innerWidth > 600 && event.target.innerWidth < 800) {
      this.pageSize = 10;
    }
    else if (event.target.innerWidth > 800 && event.target.innerWidth < 1000) {
      this.pageSize = 15;
    } else {
      this.pageSize = 5;
    }

  }

Default page size should be 5

pageSize = 5;

You can check demo here and bind it according to your requirement of screen.