How can I update a table column when a user selects an option in a dropdown list during registration?

I'm quite new to Laravel, I'm using Laravel 7 and Mysql database. I have a registration form that has a dropdown select field that pulls a list of data from mysql database. The list displays well to the user but when an option is clicked, it does not update the column in my database table once a user clicks "Register" button. The column is a foreign key because there is a one-to-one relationship between my two tables i.e "Clients" table and "Categories" table. The column name is "category_id" (This is a foreign key).

How can I update this column with an ID (from "Categories" Table) when a user selects an option?

RegisterController:

<?php

namespace App\Http\Controllers\Auth\Client;

use App\Http\Controllers\Controller;
use App\Models\User;
use App\Models\Client;
use App\Models\Category;
use App\Models\Role;
use Illuminate\Http\Request;
use Illuminate\Foundation\Auth\RegistersUsers;
use Illuminate\Support\Facades\Hash;
use Illuminate\Support\Carbon;
use Illuminate\Support\Collection;
use Illuminate\Support\Facades\Validator;
use Illuminate\Support\Facades\Auth;

class RegisterController extends Controller
{
    /*
    |--------------------------------------------------------------------------
    | Register Controller
    |--------------------------------------------------------------------------
    |
    | This controller handles the registration of new users as well as their
    | validation and creation. By default this controller uses a trait to
    | provide this functionality without requiring any additional code.
    |
    */

    use RegistersUsers;

    /**
     * Create a new controller instance.
     *
     * @return void
     */
    public function __construct()
    {
        $this->middleware('guest');
    }

    // Showing a Registration Form to User

    public function create()
    {
        
        $categories = Category::all();
      
        
        return view('auth.client.register',compact('categories'));
    }


    // Validating the User

    public function store(Request $request)
    {
        $request->validate([
            'first_name'            => 'required|string|max:255',
            'last_name'             => 'required|string|max:255',
            'telephone_number'      => 'required|digits:10',
            'email'                 => 'required|string|email|max:255|unique:users','regex:/^[\w\-\.\+]+\@[a-zA-Z0-9\.\-]+\.[a-zA-z0-9]{2,6}$/',
            'password'              => 'required|string|min:6|confirmed',
            'password_confirmation' => 'required',
        ]);

    // Creating the Client - Client is inserted in both clients and users tables

        $client = Client::create([
                    'first_name'        => $request->first_name,
                    'last_name'         => $request->last_name,
                    'telephone_number'  => $request->telephone_number,
                    'email'             => $request->email,
                    'password'          => Hash::make($request->password),
                    'category_id'       => $request->category_id,

        ]);

        $user = User::create([
                    'first_name'        => $request->first_name,
                    'last_name'         => $request->last_name,
                    'telephone_number'  => $request->telephone_number,
                    'email'             => $request->email,
                    'password'          => Hash::make($request->password),


    ]);

        $client->update(['user_id' => $user->id]);
        

        $user->email_verified_at = Carbon::now();
        $user->is_email_verified = 1;
        $user->is_active = 1;
        $user->set_password = true;

        $ownerId = $client->id;
        $ownerType = Client::class;
        $user->update(['owner_id' => $ownerId, 'owner_type' => $ownerType]);
        
        $user->assignRole('Client');

       

        return redirect('dashboard');
    }
      
}

Categories Model:

<?php

namespace App\Models;

use Eloquent as Model;
use Illuminate\Database\Eloquent\Builder;
use Illuminate\Support\Carbon;

class Category extends Model
{
    /**
     * Validation rules.
     *
     * @var array
     */
    public static $rules = [
        'name' => 'string|required|unique:categories,name',
        'color' => 'string|required',
    ];
    public $table = 'categories';
    public $fillable = [
        'name', 
        'color',
        'description'
    ];
    /**
     * The attributes that should be casted to native types.
     *
     * @var array
     */
    protected $casts = [
        'id'   => 'integer',
        'name' => 'string',
    ];
}

Relationship in my Clients Model:

/**
     * @return BelongsTo
     */
    public function category()
    {
        return $this->belongsTo(Category::class);
    }

Select field in my register blade file:

<div class="col-sm-6 col-lg-6 col-md-12">
            <label for="category">Category</label><span class="text-danger">*</span>
                <select class="custom-select" id="category_id" name="category_id">
                @foreach($categories as $category )
                <option value="{{ $category->id }}">{{ $category->name }}</option>
                @endforeach
                </select>
                </div>