UberMenu list current user’s posts as menu items

the_permalink(), the_title() by default echo their value. As you want to store that value in a variable you will need to return the values instead.
get_the_permalink(), get_the_title() are two similar functions that return the value instead of echo.

So you new code may look something like this:

$list="<li class="ubermenu-submenu ubermenu-submenu-id-122-col-0 ubermenu-submenu-type-stack">";
$list .= '<a class="ubermenu-target ubermenu-target-with-icon ubermenu-item-layout-icon_left" href="';
$list .= get_the_permalink();
$list .= '" title="';
$list .= get_the_title();
$list .= '"><span class="ubermenu-target-title ubermenu-target-text">';
$list .= get_the_title();
$list .= '</span></a></li>';