Copper and other copper alloys also harden by working. It is just that work-hardened brass is harder than work-hardened pure copper. You can find tables of work-hardening for different copper alloys online. Work-hardening is how ancient Egyptian copper blade weapons were made. I don't know their exact alloy composition, though.
The main point to be added here is that bullet jackets are made from gilding metal, which is a brass alloy. Basically, it is 5% zinc (red brass), whereas commercial cartridge cases (yellow brasses)
run from 20% to 30% typically, with 40% (Muntz metal) having been used during wartime and by a few odds and ends of manufacturers at different times.
So, being a brass alloy, bullet jackets can and do get harder than plated-on copper (Berry's, for example) or plated-on gilding alloy (Gold Dot, for example) jackets.
Does it matter how hard a bullet is? Yes. The military studied this and found during WWII that firing 30-06 armor-piercing ammunition would shorten rifle barrel life by up to about 30% as compared to jacketed lead ball ammo. These bullets have a hardened core under the jacket, so the throat and bore have to work harder to deform the surface to engrave rifling. This raises start pressure, which means the throat will endure somewhat longer exposure to peak pressures and temperatures. Shooting solid copper and bronze bullets should exhibit some of the same effects, but I don't know how much. I don't know the start pressure numbers for the AP ammo, though I would expect it to be the hardest of the lot because of the core material.
Conversely to AP, lead bullets usually exhibit lower start pressure than jacketed bullets so that the reverse effect can be observed. However, some odd exceptions come up. A number of the old revolver shooters, like Skeeter Skelton, commented that they thought lead usually raised pressure more than jacketed bullets did in hot loads, possibly due to being upset into filling out a forcing cone before having to be squeezed down again into the bore. Possibly just because they weren't allowing for cast bullets to be longer and take up more powder space than jacketed or swaged bullets. So I don't think we'll know the answer there. A military match team armorer told me he expected hardball barrels in 45 Auto to last about 25,000 rounds, while softball took forever to wear out throats and rifling. He guessed they could go 100,000 rounds, depending on the load, though that estimate, while based on great experience, is anecdotal.
There is a set of photos on page 166 in Ed McGivern's book, Fast and Fancy Revolver Shooting, that show ragged holes from 22 LR machine rest groups fired at 20 yards from a K-22 revolver with over 200,000 rounds through it. He shows groups before and after he sent it to the factory for "adjustment" that included a new cylinder stop (I had previously misremembered that it included a barrel change, but apparently that was deemed unnecessary). This is followed by the same groups repeated after the gun came back from the factory, and they don't look any different. The .22 LR runs at a similar pressure to the 45 Auto but has rifling that is only about half as tall, making the throat theoretically easier to wear down, and yet that gun clearly was not worn out at the time of the tests.