You have 2 <custom-types> opening tags and 2 closing tags </custom-types> . You should have just 1.
In the code below I removed them:
<wpml-config>
<custom-fields>
<custom-field action="copy">quantity</custom-field>
<custom-field action="translate">custom-title</custom-field>
<custom-field action="copy">weight</custom-field>
<custom-field action="translate">custom-description</custom-field>
<custom-field action="ignore">date-added</custom-field>
</custom-fields>
<custom-types>
<custom-type translate="1">project</custom-type>
<custom-type translate="1">service</custom-type>
<custom-type translate="1">staff</custom-type>
<custom-type translate="1">testimonial</custom-type>
<custom-type translate="0">slide</custom-type>
<custom-type translate="0">view</custom-type>
<custom-type translate="0">gallery</custom-type>
<custom-type translate="0">video</custom-type>
</custom-types>
<taxonomies>
<taxonomy translate="1">prj-category</taxonomy>
<taxonomy translate="0">media-tags</taxonomy>
</taxonomies>
</wpml-config>
Could you test and let me know if it works?