Can I append an array to 'formdata' in javascript?

Writing as

var formData = new FormData;
var array = ['1', '2'];
for (var i = 0; i < array.length; i++) {
    formData.append('array_php_side[]', array[i]);
}

you can receive just as normal array post/get by php.

How about this?

formdata.append('tags', JSON.stringify(tags));

... and, correspondingly, using json_decode on server to deparse it. See, the second value of FormData.append can be...

a Blob, File, or a string, if neither, the value is converted to a string

The way I see it, your tags array contains objects (@Musa is right, btw; making this_tag an Array, then assigning string properties to it makes no sense; use plain object instead), so native conversion (with toString()) won't be enough. JSON'ing should get the info through, though.

As a sidenote, I'd rewrite the property assigning block just into this:

tags.push({article: article, gender: gender, brand: brand});