When I said "battleship era" I should have said "post-dreadnought era." Dreadnought battleships, as mighty as they are, really aren't ideal for levelling a steel and concrete city. Not that they cannot, but in terms of cost effectiveness, naval cannon shells are not optimized for area damage like land artillery. Even in the Shelling of copenhagen that you linked, the english brought quite a bit of land artillery with them.
@Boris Gudenuf would have better knowledge on naval artillery in siege usage, though.
The rule of thumb in the 18th Century (or "Frigate Era" in Game Terms) was "Only a Fool would fight a Fort." Forts on land had the option of having furnaces to heat the iron shot so that any hit set fire to the wooden warship - with catastrophic results. Against a well garrisoned fortress, naval attack was just too risky. On the other hand, the Fear of what a set of Ships-of-the-Line firing 42 pound shot into the city resulted in a lot of big expensive masonry fortresses being built to cover cities - like Morro Castle in the Caribbean, for an example still nicely visible.
By the time the Dreadnaughts and Battleships came around, bombarding a big, static but Obvious target like a city was perfectly possible, but even more dangerous: mines and torpedos had been invented, and either one could take out any size ship if conditions were right. So, the German navy lost a heavy cruiser to torpedos fired from a shore installation just trying to get close enough to Oslo in Norway in 1940 to bombard it. On 18 March 1915 in what might be called a 'worse case' scenario, the French and British fleets tried to force a passage through the Dardanelles strait to get at Istanbul past two forts with about 100 guns - all but 14 of them short range, semi-obsolete pieces The forts' guns didn't do diddley-squat, but the minefields in the strait sank three (pre-dreadnaught) battleships, damaged three more, and caused a modern Battlecruiser to have to be beached to avoid sinking.
The British study of the battle concluded also that it was almost impossible to knock out the heavy guns in the forts because it took a direct hit to knock out any one gun, so had the Turks had better-trained gunners and more modern guns, it would have been much worse!
In World War Two, the most effective targets for naval gunfire were not cities (bombers did that better), but troops anywhere near the coast. Two examples out of many will serve:
In Sicily during the initial US landings in 1943 a German panzer battalion made it right onto the beach in a counterattack, and thought they had a chance to "drive the enemy back into the sea". A US Light Cruiser opened fire and virtually blew the entire battalion to bits: the Germans lost 60 tanks.The cruiser was firing at a range of less than 4000 meters, which is Point Blank for 6" naval rifled guns, and the 6" (152mm) guns on the ship could fire 2 - 4 rounds per minute each. The panzers were practically smothered in shells, and a 40 kg shell at 800 meters per second will take out any WWII tank with a single hit.
In Normandy in 1944, there's been a lot written (and shown in movies) about the US troops fighting their ay up the 5 ravines that led off Omaha Beach. In fact, they were able to start off the beach because a couple of Destroyers made a run parallel to the shore along the length of the beach and practically emptied their magazines of 5" (127mm) shells into the fortifications along the bluffs, ripping apart everything that wasn't concrete. Since most of the fortifications were field works of earth and sandbags, they were shredded and US Army engineers and infantry took out the remnants. On Utah Beach, a 15" Battleship (or possibly a Monitor, they had the same guns) hit one of the massive German concrete bunkers covering the beach. The shell didn't destroy the bunker (it's still there) but the blast blew in through all the vision ports and pulverized everything inside the bunker - including the entire crew, which was basically reduced to a thin film of Human Soup on the inside walls of the bunker.
By the second day at Normandy, elements of the 21st Panzer Division reached the coast where they thought they could flank the beachheads. Instead, they took so much naval gunfire they had to retreat immediately: being within range of destroyer 128mm, cruiser 152mm and 203mm, or Battleship 356mm and 403mm guns was simply Suicidal.
In Game Terms: Frigates (should be Ships-of-the-Line, grumble, grumble) should be able to pound cities, but not do much damage to land units, because their range is too short: even the biggest naval guns were smooth bore pieces with an effective range of less than a mile: stand off shore more than a few hundred yards, and everything not standing on the beach is out of range.
Battleships should have very little effect against cities - 20th century cities are simply too big for anything less than 2000 tons or more of high explosive (i.e., a "1000 plane raid") to do much damage, and that's more ammunition than any 4 Battleships carry! On the other hand, battleship guns can pulverize any land unit near the coast: even fortifications only delay the inevitable slightly.
Modern "Missile Cruisers" (Soviet/Russian Uvarov Class or US Ticonderoga Class is what I presume they mean) an fire lots of cruise-type missiles at a city, but have warheads that, on average, are smaller than a Battleship's 15" or 16" shells, and the cruisers don't carry any more missiles (usually a lot less, in fact) than the Battleships carried shells. The results are pinpricks to a target the size of a modern city.
What the Missile Cruisers should be able to do is take out an individual structure in the city with their GPS/TV-guidance systems: target a Factory, Barracks or Airfield and, in Game terms, Pillage it so that it's unusable until repaired. Also, their missiles have a lot more range the the Battleship's guns: 30 - 50 km versus 100s of kms.