I live in one! Melbourne, second to Sydney in Australia.
Melbourne is about to overtake/has already overtaken Sydney in population (depending on how you count) but most big corporations are headquartered in Sydney, the national media has a very Sydney-centric lense, more international flights arrive in/depart from Sydney, the Opera House is there, houses are 50% more expensive in Sydney, infrastructure projects in Sydney are prioritised, etc.
Melbourne has the larger container port, the bigger main stadium, and generally a higher ranking in global 'liveability' indices.
The other country I've lived in is Thailand, and it's harder to identify the "second city" there, as Bangkok is arguably the world's most extreme
primate city. Chiang Mai is the lead candidate, as the second largest urban area after Bangkok, but with less than a tenth of the population. It's in the far north, which is not actually a populous region outside of Chiang Mai. Khorat (3rd in population) and Khon Kaen (4th) are in the more populous Northeast region. Khorat benefits from proximity to Bangkok, Khon Kaen is more central for Northeasterners and is where big organisations have their HQs.
Now for some countries I haven't lived in:
United States: so many honourable mentions but Los Angeles still stands above the rest
Canada: Montreal over Vancouver, as Anglo-French relations/differences still have an outsized impact on politics in that country as I understand it.
Russia: undisputedly St Petersburg. The vibes I get is that Moscow and St Petersburg enjoy First World living standards while the rest of Russia has barely changed or regressed since Soviet times.
Italy: Milan, as that's where the money is.
Germany: probably Frankfurt, or Rhine-Ruhr if you count it as a single city. Actually, as I understand it, Berlin may be the capital but it's otherwise so poor that it is arguably second.
France: I believe this would be Lyon. It's the only other city in France covered by a special territorial authority.
Turkey: Ankara, second to Istanbul
New Zealand: Wellington, another case of the capital being second
Indonesia: has to be Surabaya.
Malaysia: two candidates: George Town (Penang) or Johor Bahru, just across the causeway from Singapore. I think George Town still has a higher prestige but Johor Bahru is increasingly where the money is.
Vietnam: undeniably Saigon/Ho Chi Minh City
Japan: has to be Osaka
Taiwan: Kaohsiung is generally recognised as second to Taipei. It's the main port, and arguably the cultural capital of the Taiwanese Hokkien majority, compared to Taipei which developed later and had more Mainlander influence as the capital.
China: Shanghai of course