Hi,

Please refer to the code:

<odoo>

    <template id="project_task_kanban_inherit" inherit_id="project.view_task_kanban">

        <xpath expr="//div[contains(@class,'o_kanban_content')]" position="inside">

            <div style="width: 100%; height: 50px; background-color: black;">

                <!-- Optional: Display record ID -->

                <t t-esc="record.id"/>

            </div>

        </xpath>

    </template>

</odoo>

Hope it helps.

I have found the solution:

<odoo>
    <template id="project_task_kanban_inherit" inherit_id="project.view_task_kanban">
        <xpath expr="//div[contains(@class,'o_kanban_record')]" position="inside">
            <div style="background-color: #222; color: #fff; padding: 5px; margin-bottom: 5px;">
                <div style="background-color: black; color: white; padding: 5px;">
                    Task ID: <field name="x_task_id"/>
                </div>
            </div>
        </xpath>
    </template>
</odoo>

class ProjectTask(models.Model):
    _inherit = "project.task"

    x_task_id = fields.Integer(string="Task ID", store=True)

    @api.model_create_multi
    def create(self, vals_list):
        tasks = super().create(vals_list)
        for task in tasks:
            task.x_task_id = task.id
        return tasks

You cannot access the ID because the BaseView does not include it. So in order to access it you have to create your own field and then display that. There is probably a way better way to do this. 
Also sidenote: already created tasks will not have x_task_id populated. Only new tasks. 
Im sure there is a way to actually populate already existing tasks fields with values. 

Also for some unknown reason this would not display anything before i explicitly set 'Model of the View' under Technical -> Views -> this view to 'Task'