First off, let's talk a bit about the anatomy of these tags and how they work:
Inside the bulge on the tag is some kind of device (blue) that can grip a metal pin, likely held in place by some weak springs (red). The pin (black) is pushed into the slot and the grips part/allow it to go in, but are arranged in such a way that the harder you try and pull the pin back out, the harder they grip the pin to prevent it. In store, a powerful magnet near the checkout is used to unlock the device - the bulge is inserted into the magnet and it pulls the grips (they're magnetic), overpowering the springs and drawing them away from the pin enough that the pin can then be removed. In the body of the tag (yellow area) may be a fragile vial of dye/ink that is designed to shatter and leak over the clothing, ruining them so you're discouraged from wearing or selling them, and identifying you if you've shattered it in store (in the changing rooms) and/or an electronic device detected by sensors at the door of the store to detect you walking out with the product
There are a few ways of attacking these; perhaps you can take them to a shop locally that has a similar magnet embedded in their counter, and see if they will use their magnet to unlock the device, you having shown your proof of purchase to them
If you want to remove it yourself in the same magnetic way you'll probably need some perseverance unless you can find a magnet that is the same shape as one in store, or are willing to make one (a loudspeaker magnet, drilled out.. or perhaps a few smaller magnets to place around the bulge). Some jiggling of the pin might also be required, and some patience
I've successfully removed these before with physical attack and you have a few choices:
On shoes like you have, the pin is probably put through a lace hole, so removing the head of the pin (it's the bit with the small metal part visible) by using wire cutters to nibble away at the plastic mushroom head, will mean you can get the pin out through the lace hole.
On clothing the pin would be attached through the weave of the fabric, and trying to remove it by pulling the head of the pin through the fabric may damage the fabric, even if you've reduced the size of the head of the pin
The other alternative is to gently detroy the locking end of the tag (the bulge that houses the grips). Using a pair of pliers or wire cutters you can squeeze around its circumference repeatedly, working your way round and weakening the structure of the plastic. It would be safest to engage a helper to hold the shoe and tag so that the tag is lower than the shoe - if you do manage to shatter any dye container then gravity will ensure the dye leaks downwards onto the work surface rather than the shoe. Wear gloves as a precaution, and work over an old towel. With care you'll be able to avoid damaging any dye container present
Remove enough of the bulge's outer plastic housing that you can start to pull the internals out and get the mechanism to fall apart. Even just chewing it up with the pliers may jam the grips in a position where they don't effectively hold the pin and some wiggling will remove it, but once you get the outer case of the bulge open and can see the mechanism inside you'll be able to see how to disable it
I think perhaps the quickest way would be to take an angle grinder fitted with a cutting wheel and gently cut the bulge off in its entirety, but not everyone has such a tool readily available. Pliers or wire cutters are much more likely to be something one has in the drawer, or inexpensively available in a local hardware shop
