Simple way I use:

[my param="key1=value1&key2=value2"]

in shortcode callback, just do:

parse_str( str_replace("&", "&", $attrs['param']), $array);
// var_dump( $array );

Problems:

Wordpress shortcodes have some painful restrictions in the format of the data which can be passed...

space-delimited variables:

[shortcode a="1 2"]

result: $atts=['a'='"1', 0='2"']

']' closes the shortcode:

[shortcode b=[yay]]

result: $atts=['b'='[yay']

Solution:

You can get around this by using urlencode():

[shortcode atts=a=1+2&b=%5Byay%5D]

parse it like so:

parse_string($atts['atts'],$atts);

result: $atts=['a'=>'1 2', b=>'[yay]']

That will give you the array passing you want.

Now as for building your menu(s):

function internal_menu($atts) {
  // allow passing + and ] in the text of the links:
  parse_string($atts["links"],$links);

  // the defaults, verbatim from the question:
  if (!count($links)) $links=[
    'href1' => '#jl1',
    'href2' => '#jl2',
    'href3' => '#jl3',
    'href4' => '#jl4',
  ];

  foreach ($links as $text=>$href) $ul=."<li><a href=\"$href\">$text</a></li>";

  return '<div id="internalPageMenu"><ul>'.$ul.'</ul></div>';
}

add_shortcode('internal-menu', 'internal_menu');

//menu target

function internal_menu_target($atts) {
  // allow passing + and ] in the text:
  if (@$atts[text]) $atts['text']) = urldecode($atts['text']);

  // the defaults, verbatim from the question:
  $atts=array($atts)+['text'=>'','id'=>'jl1'];

  return '<h3 id="' . $link['id'] . '">' . $link['text'] . '</h3>';
}

add_shortcode('internal-menu-target', 'internal_menu_target');

and then feed it like so:

[internal-menu links=First+Link=#jl1&Second+Link=#jl2&Google=https://google.com]

[internal-menu-target text=Section+1 id=jl1]

etc.

BTW extract() is perfectly fine if you use it responsibly:

/* concatenates 3 words of wisdom into a polite policy direction
 *
 * @param $attr: hash of function args
 *          foo = the first word (Kindly)
 *          bar = the second word (ignore)
 *          baz = the third word (totalitarians)
 */

function excellent_policy($attr){
  $defaults=['foo'=>'Kindly', 'bar'=>'ignore', 'baz'=>'totalitarians'];
  extract((array)array_intersect_key($attr,$defaults)+$defaults);
  echo "$foo $bar $baz!";
}

This imports $foo, $bar, and $baz from $attr into local scope in a readable and predictable manner, provides defaults for those variables if they were not passed, and prevents the creation of any unexpected variables.

There's good ways and bad ways to use language features. Forbidding everyone from using a language feature because someone might use it poorly is like forbidding everyone from breathing because someone might try to inhale Jello.

I have an alternative way to do this in $content of short code instead.

This solution increases the readability/relation of your data. But it might not be a good way if you want to use your $content to do other things too.

shortcode used is like...

// in wordpress editor or your *.php file

[testimonial_carousel_list]
[
  {
    "message": "Lorem ipsum dolor sit amet.",
    "avatar": "https://via.placeholder.com/150",
    "full_name": "John Doe"
  },
  {
    "message": "Vivamus vel ornare purus, in faucibus tellus.",
    "avatar": "https://via.placeholder.com/150",
    "full_name": "Jane Smith"
  },
  {
    "message": "Morbi tristique augue vel mi ornare, sit amet viverra lectus semper.",
    "avatar": "https://via.placeholder.com/150",
    "full_name": "David Lee"
  }
]
[/testimonial_carousel_list]

And here how to handle it intro your shortcode function

// it might be in functions.php of your theme

function testimonial_carousel_list_shortcode($_atts, $content = null) {
  $items = json_decode($content);

  // ...rest of your code whatever you what to do with array $items
}
add_shortcode('testimonial_carousel_list', 'testimonial_carousel_list_shortcode');

// this is required if you want to json_decode() on your $content.
// because wptexturize will change your `'` or `"` to `‘` or `“` it will made json_decode got error.
function shortcodes_to_exempt_from_wptexturize( $shortcodes ) {
    $shortcodes[] = 'testimonial_carousel_list';
    return $shortcodes;
}
add_filter('no_texturize_shortcodes', 'shortcodes_to_exempt_from_wptexturize');

I like your approach @Akegvd and was able to alter it so that it'll also work inside Gutenberg.

In Gutenberg plain text characters are replaced with formatted entities through wptexturize(). According to the documentation you can avoid this transformation easily with putting the relevant content inside special tags, e.g. <code>. With this you can avoid your json information, i.e. the double quotes being converted and furthermore use the closing tag </code> to separate the json from the rest of your shortcode's $content.

This provides a good solution to have a good readability and clean structure of the json data and also use the $content for other purposes.

Simply use the shortcode with <code> tag

// in Wordpress Gutenberg shortcode block write

[testimonial_carousel_list] 
<code>
[   
  {
    "message": "Lorem ipsum dolor sit amet.",
    "avatar": "https://via.placeholder.com/150",
    "full_name": "John Doe"   
  },   
  {
    "message": "Vivamus vel ornare purus, in faucibus tellus.",
    "avatar": "https://via.placeholder.com/150",
    "full_name": "Jane Smith"   
  },
  {
    "message": "Morbi tristique augue vel mi ornare, sit amet viverra lectus semper.",
    "avatar": "https://via.placeholder.com/150",
    "full_name": "David Lee"   
  }
] 
</code>
SOME OTHER CONTENT
[/testimonial_carousel_list]

and write the following in your functions.php

// it might be in functions.php of your theme

function testimonial_carousel_list_shortcode($atts, $content = null) {
  
    $atts = shortcode_atts( array(      
      'somedefaultkey' => 'val1',
      'otherdefaultkey' => 'val2'), 
       $atts ); 
            
    $temp=explode('</code>', $content);
    $json_str=$temp[0];
    $content=$temp[1];

    // strip automatically added html tags e.g. <br>    
    $json_str=strip_tags($json_str); 
    $items = json_decode($json_str, true); # set true to get accessible $items array

    // Loop through json_entries
    foreach($items as $value){
      // do something
    }
     // ...rest of code
}
add_shortcode('testimonial_carousel_list', 'testimonial_carousel_list_shortcode');

Hope this helps @Thomas D.