How to get posts that contain multiple terms from multiple taxonomies?

Unfortunately, you can't just pass an array of term ids into a WP function and get all those posts returned if the term ids are of different taxonomies. You'll have to do it the hard way with tax-query key passed in get_posts(). Fortunately, even though it is the hard way, you have complete control over the returned posts list, and a well designed tax-query can give you virtually any customization you can think of.

In your example tax-query arguments, you are calling category__and and tag__and. It would be better to call all desired taxonomies in a single tax-query so that you can design the result according to your needs.

The way I would do this is build the arrays within the tax-query separately and call them by variable later, like this:

$get_posts_args['tax_query'] = array(
            'relation' => 'AND',
            $tag_array,
            $cat_array,
            $custax1_array,
            $custax2_array, ); }

The first benefit is that if any of these arrays are empty, they are simply ignored and the query runs anyway. The second benefit is that you can set further operator logic within these arrays. For example, $tag_array may look like this:

$tag_array = array(
        'taxonomy' => 'post_tag',
        'field'    => 'term_id',
        'operator' => 'AND',
        'terms'    => $tag_ids,); }

The operator key allows a few options that work within the taxonomy, while the relation key in the outer array works on the logic between the taxonomies. If you design these arrays programmatically, it becomes trivial to change the logic as needed for both circumstances for all taxonomies. For hierarchical taxonomies, you probably want to consider your preferences for the include_children key.

The somewhat unique problem of having an integer array containing term ids from unknown taxonomies is pretty easily resolved with:

$tag_ids = get_terms(array(
     'include'=>$term_ids_list,
     'taxonomy'=>'post_tag',
     'fields'=>'ids'));

If $term_ids_list is your integer array of term ids, then calling get_terms() for a specific taxonomy will return only the terms within that taxonomy. This gives you $tag_ids as an integer array consisting only of term ids belonging to the tag taxonomy. You'd simply change the keys accordingly for the taxonomy you want to target.

In hindsight for resolving this problem, I find that it is better to work with the taxonomies separately, even if they will be presented to the user as a single list. Presumably, we make taxonomies to separate kinds and types of content. Not keeping them separate undoes whatever purpose you originally had for making the separation in the first place. In my case, I wanted to have small urls, so decided that a single url variable like ?terms=1,2,3,4,5 would be better than ?cats=1,2&tags=3,4&custax1=5. Indeed, the url is smaller, but upon reload the taxonomy key is lost. In this specific case however, it's not too difficult to put them back together.

I suggest that that @SallyCJ's approach makes much more sense if you can do it this way: simply break down wherever these ID's get written to store the tax names at that point.

However it seems like figuring out which Taxonomy each Term ID is in and then using that to generate the tax_query part of the WP_Query args isn't so hard. get_term allows you to find the taxonomy name from a term ID, so with that you're half way there.

This is untested code, some of it may contain errors and uses your $args as a base so you may need to double check the structure of the tax_query part, but it contains all the pieces you could use to build the code if you wanted to do it this way:

$arrTermIDs = [ 1,2,3,4,5 ];
$arrTaxAndTerms = []; // make a 2d array with the term ID's broken down by tax name

foreach ($arrTermIDs as $termID) {
    $term = get_term($termID);
    if (!in_array($term->taxonomy, $arrTaxAndTerms)) 
        $arrTaxAndTerms[$term->taxonomy] = [] ; // create array for this tax if we don't have one already
    
    $arrTaxAndTerms[$term->taxonomy][] = $termID; // add this term id to array for this tax
}

// Now we have all the info we need to build the tax_query items for each taxonomy 
$arrTaxQueryItems = [];

foreach($arrTaxAndTerms as $taxName => $termIDs) {
    $arrTaxQueryItems[] = 
        array(
              'taxonomy' => $taxName,
              'field'    => 'term_id',
              'terms'    => $termIDs
        );
}

// Put that together into your $args:
$args = array(
    'numberposts' => -1,
    'post_type' => array('post'),
    'post_status' => 'publish',
    // any other conditions you want here
    'tax_query' => array(
         'relation' => 'AND',
         $arrTaxQueryItems
    )
)