Simple solve.  Create elems with flags.

lets create array of elements with default settings. let be `true` by. default. And create 1 method for change state of elems.

...

    public elems = {
    pid: true,
    doc: true,
    rn: true
    }
    ....
    
     toggle(elem, state) {
       this.submitOpt = state;
       elem = state;
     }

in template will use *ngIf and click change state of flags elem


    <div class="row pl-15 pr-15">
      <div id="pid-container" class="row" *ngIf="elem.pid">
        <div class="col-3 pl-0">
          <mat-form-field>
            <input matInput id="police-id" [(ngModel)]="policyId">
          </mat-form-field>
        </div>
        <div class="col-1 align-self-center">
          <button mat-raised-button class="main" (click)="toggle(elem.pid,true)">
            Search
          </button>
        </div>
        <div *ngIf="policyId != '' && submitOpt">
          <app-policies-list [searchValue]="policyId"></app-policies-list>
            <div class="row">
              <div class="col text-align-right">
                <button mat-raised-button class="main" (click)="toggle(elem.pid,false)">
                  Back
                </button>
              </div>
            </div>
        </div>
      </div>
    <div id="rn-container" class="row" *ngIf="elem.rn">
      <div class="col-3 pl-0">
        <mat-form-field appearance="outline" class="formfield">
          <input matInput id="request-number" [(ngModel)]="requestNumber">
        </mat-form-field>
      </div>
      <div class="col-1 align-self-center">
        <button mat-raised-button class="main" (click)="toggle(elem.rn, true)"
          Search
        </button>
      </div>
      <div *ngIf="requestNumber != '' && submitOpt">
        <app-policies-list [searchValue]="policyId"></app-policies-list>
          <div class="row">
            <div class="col text-align-right">
              <button mat-raised-button class="main" (click)="toggle(elem.rn, false)">
                Back 
              </button>
            </div>
          </div>
      </div>
    </div>
    <div class="row" id="doc-container" *ngIf="elem.doc">
      <div class="col-3 pl-0">
        <mat-form-field appearance="outline" class="formfield">
          <input matInput id="documentation" [(ngModel)]="documentationNumber">
        </mat-form-field>
      </div>
      <div class="col-1 align-self-center">
        <button mat-raised-button class="main" (click)="toggle(elem.doc,true)">
          Search
        </button>
      </div>
      <div *ngIf="documentationNumber != '' && submitOpt">
        <app-policies-list [searchValue]="policyId"></app-policies-list>
        <div class="row">
          <div class="col text-align-right">
            <button mat-raised-button class="main" (click)="toggle(elem.doc,false)">
              Back
            </button>
          </div>
        </div>
      </div>
    </div>

You shouldn't use classes. General  is use variables in Angular. In fact, we got rid of two big methods and moved the display logic to the template