Fixed! This was a dumb mistake! Removed [] from $value
if ( is_user_logged_in() ) {
$user_id = get_current_user_id();
$field_key = 'field_5bdc8d3c953f2';
$value = get_field($field_key, 'user_'.$user_id);
$new_row = array('name' => $dogsname, 'sex' => $sex, 'rda' => $rda);
array_unshift( $value, $new_row );
update_field($field_key, $value, 'user_'.$user_id);
}