Use PHP's array_filter function with a callback.

$new = array_filter($arr, function ($var) {
    return ($var['name'] == 'CarEnquiry');
});

Edit: If it needs to be interchangeable, you can modify the code slightly:

$filterBy = 'CarEnquiry'; // or Finance etc.

$new = array_filter($arr, function (filterBy) {
    return ($var['name'] == $filterBy);
});

Your idea of using a built-in function like array_filter is a very good one; PHP has lots of these that can make your life easier.

Specifically, array_filter accepts a callback that you can use to customize the filtering logic. This would work:

$filtered = array_filter($array, function($el) { return !empty($el['mob']); });

Here the callback is supplied as an anonymous function.

I decided to rewrite my answer to accommodate both filtering and sorting. I took a heavily object oriented approach to solving this problem, which I will detail below.

You can see all of this code in action at this ideone.com live demonstration.

The first thing I did was define two interfaces.

interface Filter {
    public function filter($item);
}

interface Comparator {
    public function compare($a, $b);
}

As their names suggest, Filter is used for filtering, and Comparator is used for comparing.

Next, I defined three concrete classes that implements these interfaces, and accomplish what I wanted.

First is KeyComparator. This class simply compares the key of one element to the key of another element.

class KeyComparator implements Comparator {
    protected $direction;
    protected $transform;
    protected $key;

    public function __construct($key, $direction = SortDirection::Ascending, $transform = null) {
        $this->key = $key;
        $this->direction = $direction;
        $this->transform = $transform;
    }

    public function compare($a, $b) {
        $a = $a[$this->key];
        $b = $b[$this->key];

        if ($this->transform) {
            $a = $this->transform($a);
            $b = $this->transform($b);
        }

        return $a === $b ? 0 : (($a > $b ? 1 : -1) * $this->direction);
    }
}

You can specify a sort direction, as well as a transformation to be done to each element before they are compared. I defined a helped class to encapsulate my SortDirection values.

class SortDirection {
    const Ascending = 1;
    const Descending = -1;
}

Next, I defined MultipleKeyComparator which takes multiple KeyComparator instances, and uses them to compare two arrays against each other. The order in which they are added to the MultipleKeyComparator is the order of precedence.

class MultipleKeyComparator implements Comparator {
    protected $keys;

    public function __construct($keys) {
        $this->keys = $keys;
    }

    public function compare($a, $b) {
        $result = 0;

        foreach ($this->keys as $comparator) {
            if ($comparator instanceof KeyComparator) {
                $result = $comparator->compare($a, $b);

                if ($result !== 0) return $result;
            }
        }

        return $result;
    }
}

Finally, I created MultipleKeyValueFilter which is meant to filter an array based on an array of key/value pairs:

class MultipleKeyValueFilter implements Filter {
    protected $kvPairs;

    public function __construct($kvPairs) {
        $this->kvPairs = $kvPairs;
    }

    public function filter($item) {
        $result = true;

        foreach ($this->kvPairs as $key => $value) {
            if ($item[$key] !== $value)
                $result &= false;
        }

        return $result;
    }
}

Now, given the input array (Notice I rearranged them a bit to make the sorting obvious):

$array = array (
    '1' => array ('type' => 'blah2', 'category' => 'cat2', 'exp_range' => 'this_week' ),
    '2' => array ('type' => 'blah1', 'category' => 'cat1', 'exp_range' => 'this_week' ),
    '3' => array ('type' => 'blah1', 'category' => 'cat2', 'exp_range' => 'next_week' ),
    '4' => array ('type' => 'blah1', 'category' => 'cat1', 'exp_range' => 'next_week' )
);

Sorting can be achieved by doing the following:

$comparator = new MultipleKeyComparator(array(
    new KeyComparator('type'),
    new KeyComparator('exp_range')
));

usort($array, array($comparator, 'compare'));

echo "Sorted by multiple fields\n";
print_r($array);

Filtering can be achieved by doing the following:

$filter = new MultipleKeyValueFilter(array(
    'type' => 'blah1'
));

echo "Filtered by multiple fields\n";
print_r(array_filter($array, array($filter, 'filter')));

At this point I've given you a great deal of code. I'd suggest that your next step is to combine these two pieces into a single class. This single class would then apply both filtering and sorting together.

Might be more complex than needed but maybe a combination:

$array = array_map(function($v) {
                   return array_filter($v, function($v) {
                                           return $v['foo'] === 'bar'; }); }, $array);

This should also work:

foreach($array as &$value) {
    $value = array_filter($value, function($v) { return $v['foo'] === 'bar'; });
}

You can use a recursive function for deeper or unknown nesting:

function filterNotSomething(&$array, $key, $val) {
    foreach($array as $k => $v) {            
        if(isset($v[$key]) && $v[$key] !== $val) {
            unset($array[$k]);
        }
        elseif(is_array($v)) {
            filterNotSomething($array[$k], $key, $val);
        }
        if(empty($array[$k])) {
            unset($array[$k]);
        }
    }
}

filterNotSomething($array, 'foo', 'bar');

The hint is already there, to filter, use array_filter.

Don't forget to use use keyword to import your criteria.

Example:

$newArray = array_filter($myArray, function($e) use ($filter){
                                                //    ^ import criteria
    return in_array($e['name'], $filter);
});

Of course you can simply loop any given input and check if a given element matches any criteria you implement:

<?php
$input = [
    [
        'id' => 2135,
        'first_name' => 'John',
        'last_name' => 'Doe',
        'gender' => 'male'
    ],
    [
        'id' => 3245,
        'first_name' => 'Sally',
        'last_name' => 'Smith',
        'gender' => 'female'
    ],
    [
        'id' => 5342,
        'first_name' => 'Jane',
        'last_name' => 'Doe',
        'gender' => 'female'
    ],
    [
        'id' => 5623,
        'first_name' => 'Peter',
        'last_name' => 'Doe',
        'gender' => 'male'
    ],
    [
        'id' => 7216,
        'first_name' => 'Mike',
        'last_name' => 'Lill',
        'gender' => 'male'
    ]
];

$needles = ['Doe', 'male'];
$output = [];
array_walk($input, function($element) use ($needles, &$output) {
    $matches = true;
    foreach ($needles as $needle) {
        if (!in_array($needle, $element)) {
            $matches = false;
        }
    }
    if ($matches) {
        $output[] = $element;
    }
});

print_r($output);

The obvious output is:

Array
(
    [0] => Array
        (
            [id] => 2135
            [first_name] => John
            [last_name] => Doe
            [gender] => male
        )

    [1] => Array
        (
            [id] => 5623
            [first_name] => Peter
            [last_name] => Doe
            [gender] => male
        )

)