It's bad software design in the Matrix or an exploit. :)
Something like this is actually possible in some games following practices I'd consider bad game design (partially intended and unavoidable though).
I had to immediately think of Neverwinter Nights 2, a RPG with an optional online mode, where you might bring your own character to any online world that allows local characters (i.e. characters not exclusively stored on the server).
Basically, MMORPGs follow the simple rule The client is in the hands of the enemy. As such, they will only allow you to take a specific set of parameters with you when you create your avatar. The client can't tell the server how much damage they do, which skills they've unlocked, etc. For example, you might pick a race, class, skin color, hair style, hair color, etc. Sounds a bit familiar? Kinda what they do when entering the Matrix.
However, NWN2 (and some other games) go a bit further: They allow the client to dictate the server what they're wearing or carrying in their inventory, i.e. uploading this when you establish a connection. Now it's getting interesting. And there are also some parallels to The Matrix once again:
Players indeed setup small maps using the map editor that work pretty much like the armory in the movie. You've got one or several rooms filled with all kind of items, like weapons, clothes, potions, etc.
So what they do is enter such a modified map, equip the stuff, then save and exit, entering the actual multiplayer world.
They do the very same thing in the movie: They enter their construct/armory, equip weapons, and then enter the actual Matrix. Of course, that inventory is not really saved with their "savegame", but that additional data or payload is obviously sent when setting up their connection. How it's actually done isn't revealed, but there are multiple possibilities.
Once again, from a real world example, when opening a web page, your browser submits additional data (e.g. cookies), that the script running on the server will be able to load/interpret. If you've heard of the Shellshock vulnerability earlier this year, this is an attack most likely pretty close to what they do when entering the Matrix:
You send a request to the server together ("I want to enter the Matrix.") with some additional payload that's not used normally (i.e. the weapon data/inventory). Due to some bug or security issue, they then get the server to actually retrieve and use that data ("materializing" their weapons).