The Udaloys are not "frigates".
Fregat is their official Russian name much like "Grisha"is called
Albatros and "Krivak" is
Burevestnik.
Right now there are two main systems of ship classification - NATO and Warsaw Pact. China's current surface fleet follows NATO. Russia is slowly transitioning to a NATO-like classification but many ships are of Soviet design which is why they use old classifications. The fundamental difference is the result of different doctrinal requirements. NATO had to counter Soviet threat while retaining sea control. Warsaw Pact protected its coastal zone and countered NATO power projection. NATO was an alliance of established maritime powers (US, UK, France, Italy, Netherlands) while Warsaw Pact was a alliance of land powers. These two are different strategies and require different types of ships.
NATO
Contemporary NATO classification aligns most with West European navy classifications developed in the 1960s-80s and is closest to Royal Navy nomenclature but it includes elements of post-1975 USN classification due to the role and size of US Navy.
- cruiser - large surface combatant, blue-water operation, anti-air warfare, capable of task force command (carrier and landing escort, convoy escort)
- destroyer - large surface combatant, blue-water operation, anti-air or anti-submarine warfare capable of small task force command (convoy escort, small flotilla)
- frigate- small surface combatant, blue-water operation, anti-submarine warfare and general patrol duties, incapable of command
- corvette - small surface combatant, green-water operation, general patrol duties, incapable of command
This is how NATO navies operated in the 1980s which is the defining moment for NATO naval structure because it emerges as response to a very specific doctrine - countering Soviet threat. The Cold War starts in the 1950s and until the 1970s the navies largely use ships from WW2 or WW2-era designs. In the 1970s both the US Navy and European navies undergo a shift in fleet structure and ship design which is geared toward countering Soviet threat.
- cruiser - Ticonderoga,
- destroyer (AAW) - Type 42, Tromp, Cassard
- destroyer (ASW) - Spruance, Georges Leygues,
- frigate (ASW+GP) - Oliver Hazard Perry, Knox, Kortenaer, Bremen, Type 23, Type 22
- corvette (GP) - Joao Belo, Descubierta, D'Estienne D'Orves
What happens in the 1990s is that the fleet structure and ship design stagnates much like both stagnated in the post-WW2 era but the ships "grow" to accomodate for new capabilities and technologies. Because the growth occurs like inflation of the universe - across all classes - the shift in nomenclature occurs naturally. This leads to a lot of confusion because with the growth of ship size and reduction of fleet size the capabilities are merged. Right now you have "specialized" and "not specialized" ships. Specialized AAW ships have dedicated radars (AEGIS, APAR+SMART-L) while not specialized ships have just anti-air missiles with range exceeding the old Sea Sparrow. Specialized ASW ships are designed to reduce noise generation and are equipped with the full complement of sonars (towed array + hull sonar) and have facilities for sustaining ASW helicopters while not specialized ships have just basic hull sonars and torpedos and a basic helicopter.
Germany, Netherlands and Denmark have anti-air frigates even though technically they should be called destroyers. They are called a frigate because they are derived from frigate design and they operate in the same manner as frigates do- they are commanded by the same officer rank, have the same crew size etc. France and Italy operate the Horizon/Orizzonte class which is currently classified as a frigate but initially was a destroyer.
Warsaw Pact
When the Cold War started Soviets quickly abandoned unrealistic plans devised under Stalin which involved building large battleships and heavy cruisers. They focused on securing coastal security to protect themselves from NATO power projection from the sea. To that purpose they followed the doctrine known as "jeune ecole" or German strategy from WW2 which emphasized submarines and small attack vessels as well as large number of patrol and ASW ships. Their ships were not defined by their seaworthiness and role at sea like NATO, but by their primary combat mission. Hence "large anti-submarine ship" or "small anti-submarine ship" or "guard ship"
- Большие противолодочные корабли - large anti-submarine ships (Udaloy, Kashin, Kara)
Малые противолодочные корабли - small anti-submarine ships (Grisha, Pauk)
Сторожевые корабли - guarding ships (Krivak, Neutrashimy, Stereghuschy, Gorshkov)
Эскадренные миноносцы - squadron torpedo ships (Sovremenny)
As you can see the first category has a "destroyers" and a "cruiser", the second has corvettes of various sizes, the third category includes general purpose seagoing ships and the fourth is really a large anti-surface ship.
The Soviet navy also had proper cruisers and those were limited to three classes of ships armed in guided missiles - Kresta I (Berkut), Slava (Atlant) and Kirov (Orlan).
Right now Russia is changing the classification of the ships to be able to better market it for export. This is why most "guarding ships" become frigates, even though Steregushchy is called a corvette (and should be - according to NATO classification due to its low autonomy) while "small missile ships" are called corvettes.
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If anyone is interested in Soviet and Russian naval vessels I recommend this great website:
It's in Russian but you can choose English version (bottom of the menu to the left) - the two versions are identical in terms of information.
The reason why most navies - including China - are moving to NATO classification is because this classification describes a fleet structure that is appropriate to a maritime power. It is much more oriented to power projection rather than just narrow combat mission like the Warsaw Pact navies. The names are just convenience. It's the ships, their roles and design that matter.
China returns to its place as a maritime power. It is only logical that it will use a system that is developed by and for maritime powers.