As of Bootstrap 5.1.3 the background color component is not automatically generated by

$utilities-bg-colors: map-loop($utilities-colors, rgba-css-var, "$key", "bg");

In the manual it is written,

Unfortunately at this time, not every component utilizes this Sass map. Future updates will strive to improve upon this. Until then, plan on making use of the ${color} variables and this Sass map.

Here is one way of manually adding .bg-* classes in Bootstrap 5.1.3. Compiled with Parcel V 2.3.2

/*Bootstrap core imports - adjust for your case*/
@import "~bootstrap/scss/functions";
@import "~bootstrap/scss/variables";
@import "~bootstrap/scss/utilities";
@import "~bootstrap/scss/mixins";


$custom-theme-colors:(
  "custom-color": #8bd0da,
);

$theme-colors: map-merge($theme-colors, $custom-theme-colors);
$theme-colors-rgb: map-loop($theme-colors, to-rgb, "$value");
$utilities-colors: map-merge($utilities-colors, $theme-colors-rgb);
$utilities-text-colors: map-loop($utilities-colors, rgba-css-var, "$key", "text");
$utilities-bg-colors: map-loop($utilities-colors, rgba-css-var, "$key", "bg");

// Adjust path to bootstrap for your case
@import "~bootstrap/scss/bootstrap";

// .bg classes not automatically generated. As an example, manually add
// .bg-custom-color

.bg-custom-color{background-color: var(--bs-custom-color);};

Example html

<div class="container border mb-4 bg-primary">
    Background: bg-primary
</div>

<div class="container border mb-4 bg-custom-color">
    <code>bg-custom-color</code>
</div>

<div class="container mb-4">    
    <div class="btn btn-custom-color"><code>btn-custom-color</code></div>
</div>

<div class="container mb-4">
    <div class="alert-custom-color">alert-custom-color</div>
</div>

<div class="container mb-4">
    <ul class="list-group">
        <li class="list-group-item-custom-color"><code>list-group-item-custom-color</code></li>
    </ul>
</div>

The page render as shown below