Saving array of data to database using wordpress settings API

It would be good if you added your code to see more or less what you want to do.

You can not pass it that way, because when using the Settings API you must add fields of a single value and are registered with the function add_settings_field().

I do not understand very well what you want to do but if you need to create a two-dimensional or multidimensional array using the value in the name attribute in the following way with its options to save:

// You add the corresponding labels, if I'm not wrong, you should have it.
<select name="my_plugin_options[mode][low]">
    <option value="low">Low</option>
    // or true
    <option value="true">Low</option>
</select>

<select name="my_plugin_options[mode][high]">
    <option value="high">High</option>
    // or true
    <option value="true">High</option>
</select>  

You would obtain the value in the following way:

<?php

    $my_plugin_options = get_option( 'my_plugin_options' );
    var_dump( $my_plugin_options[ 'mode' ] );
    // array( 'high' => 'high' ) or array( 'high' => 'true' )
    // or
    // array( 'low' => 'low' ) or array( 'low' => 'true' )

    echo $my_plugin_options[ 'mode' ][ 'high' ]; // 'high' or 'true'
    echo $my_plugin_options[ 'mode' ][ 'low' ]; // 'low' or 'true'

    // Example:

    if( $my_plugin_options[ 'mode' ][ 'high' ] == 'high' ) { // Actions }
    if( $my_plugin_options[ 'mode' ][ 'high' ] == 'true' ) { // Actions }
    // or
    if( $my_plugin_options[ 'mode' ][ 'low' ] == 'low' )  { // Actions }
    if( $my_plugin_options[ 'mode' ][ 'low' ] == 'true' ) { // Actions }

?>  

If you could add your test code, it would be better for me to help you.

or

<select name="my_plugin_options[mode]">
    <option value="low">Low</option>
    <option value="high">High</option>
</select>

<?php

    $my_plugin_options = get_option( 'my_plugin_options' );
    var_dump( $my_plugin_options[ 'mode' ] ); // 'high' or 'low'

    echo $my_plugin_options[ 'mode' ]; // 'low' or 'high'

    // Example:

    if( $my_plugin_options[ 'mode' ] == 'high' ) { // Actions }
    if( $my_plugin_options[ 'mode' ] == 'low' )  { // Actions }

I recommend using the Options API with the function add_option() or update_option().

Example:

<?php  

// You receive the data and save it
update_option( 'my_plugin_options', $_POST[ 'mode' ] );

$my_plugin_options = get_option( 'my_plugin_options' );

echo $my_plugin_options[ 'mode' ]; // 'low' or 'high'

If you could add your test code, it would be better for me to help you.