this.registerForm.controls['email'].setErrors({ invalid: true, valid: false })

export class RegistrationComponent implements OnInit {

  ...

  constructor(...) {}

  ngOnInit() {
    this.registerForm = this.fb.group({
      email: ['', [FormValidationService.emailValidator]],
      password: this.fb.group({
        pwd: ['', [FormValidationService.passwordValidator]],
        confirm: ['', [FormValidationService.passwordValidator]],
      }, {validator: FormValidationService.confirmValidator('pwd', 'confirm')}),
      username: ['', [FormValidationService.shortStringValidator]],
    })
  }

  register() {
    this.spinnerShow = !this.spinnerShow
    const regData = this.registerForm.value
    regData.confirm = this.registerForm.value.password.confirm
    regData.password = this.registerForm.value.password.pwd

    this.auth.registration(regData).subscribe(
      (res: any) => {
        if (res.error) {
          this.spinnerShow = !this.spinnerShow
          const messages = res.messages
          for (const key in messages) {
            if (messages.hasOwnProperty(key)) {
              if (key === 'both') {
                this.bothErrMsg = messages[key]
                this.registerForm.controls['email'].setErrors({ notUnique: true })
              } else {
                this.registerForm.controls[key].setErrors({ errMsg: messages[key] })
              }
            }
          }
        } else {
          this.lc.setItem('user', res.user)
          this.matDialogRef.close()
        }
      },
      error => {
        console.log(error) // TODO
        this.spinnerShow = !this.spinnerShow
      }
    )
  }
}

<div class="container">
  <h3>Register new user.</h3>
  <form [formGroup]="registerForm" novalidate>
    <mat-form-field>
      <input matInput formControlName="email" placeholder="Email" type="email" required>
      <mat-error *ngIf="registerForm.controls.email.hasError('errMsg') && registerForm.controls.email.touched">
        {{ registerForm.get('email').getError('errMsg') }}
      </mat-error>
    </mat-form-field>

    <div class="error" *ngIf="bothErrMsg !== undefined">{{ bothErrMsg }}</div>

    <button mat-raised-button color="primary" [disabled]="registerForm.invalid == true" (click)="register()">Sign Up</button>
  </form>
</div>

<form [formGroup]="registerForm">
    <mat-form-field>
        <input matInput placeholder="Email" formControlName="email">
        <mat-error *ngIf="email.dirty && email.hasError('notUnique')">This email is already taken!</mat-error>
    </mat-form-field>
</form>

public registerForm = new FormGroup({
    email: new FormControl(null, {
        validators: [Validators.required, Validators.email]
    })
});

public get email(): AbstractControl {
    return this.registerForm.get('email')!;
}

public register(): void {
    this.auth.registration(this.registerForm.value).subscribe((res) => {
        this.email.setErrors({ notUnique: true });
    });
}