(Solid) tip "tinners" are almost always ammonium phosphate with some tin (or tin-lead) bits mixed in. They are moderately aggressive at cleaning oxides off, so shouldn't be used constantly, but only when needed. The best tip tinner is your solder--tips should always be coated with a small amount of solder.
If some of the oxides are just sticking really well, you could try to mildly abrade them on a brass sponge, copper braid, or similar, but you can't be too hard or you will damage the iron plating (good tips are typically copper core, plated with iron, then chromium everywhere but the working area).
Practice good tip care. I use a Hakko iron and tips at work and mine has lasted about a year (moderately light use, maybe 5-6 projects). Put a good amount of solder on the tip when storing it or leaving it idle for any length of time; don't, for instance, wipe it off, put it in the holder, then leave it on over lunch.
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This looks to me like a combination of too high heat and crappy tip coating. Unfortunately this is common with unregulated soldering irons. They usually get much hotter than a regulated iron because they always put out the same power, and that power has to be enough to solder against a ground plane or something else that acts like a heat sink. A regulated iron would just crank up the power when needed, but a unregulated iron is stuck with a fixed power, so gets too hot, sometimes way too hot, when just sitting in the holder. I've seen a unregulated 25W soldering iron get hot enough to visibly glow in a dark room.
The excessive heat can not only damage whatever you're soldering, but it speeds up the oxidation reaction that eventually ruins tips. Solder oxidizes fairly quickly, even at normal soldering temperatures. You may have tinned the tip regularly, but I'm sure that didn't last long. Then the tip coating starts to oxidize. Once that's worn off, you're left with copper. Copper is actually eaten away by soldering itself. It migrates into the tin/lead mix, which is one reason solder works so well on copper.
People that buy unregulated irons are shopping on immediate price, so manufacturers make them as cheap as possible. That probably includes the cheapest possible tip coating that looks good out of the box.
So in short, it looks like you got what you paid for.
Best Answer
Steel is typically a harder metal than the plating on soldering iron tips. So while coarse steel wool might clean a tip, it would also quickly destroy the plating, thus it should only be used "in an emergency" or with a cheap tip you are willing to soon replace.
The metal sponges arguably appropriate are made of softer materials, such as brass. And they really should be brass all the way through, not some sort of copper plating on a steel core, as you might source at the dollar/kitchen store.
But even with a soft material, abrasion counts - something like a cellulose sponge will also eventually wear a tip. And so will sitting hot with solder and flux on it.
Tips are wear items. The less they are "cleaned" and the less time they sit hot, the longer they last. But cleaning is of course necessary, especially for modern fine pitch soldering.
The most modern professional solder stations have high power heaters and very tight control loops, so they can quickly go into a reduced temperature tip-saving mode when idle, and heat right back up again in the time the operator picks up the iron from the stand and applies to the work.