Chinese officials typically talk like that when the topic is Taiwan or something else where the US opposes them, e.g. the AUKUS deal and what not.
This can even be stuff where the opposition isn't that obvious e.g. "China sees ‘Cold War mentality’ in US-Vietnam pact, Vietnamese disagree"
China’s foreign ministry spokesperson Mao Ning advised Washington to discard its “hegemonic and Cold War mentality” when asked about Trong and Biden signing up to a comprehensive strategic partnership equal to Beijing’s own relations with Vietnam.
Sometimes such critiques go hand in hand with a call/emphasis on an Asian identity, e.g.
In a thinly veiled swipe at the US, Wang said: “For geopolitical reasons, a certain major power outside the region has deliberately exaggerated our ideological differences, organized all kinds of cliques for the purpose of excluding others, tried to replace cooperation with confrontation, and unity with division," transcript of his speech released by China's foreign ministry read.
During his conversation with the guests attending the forum, Wang said: "Europeans and Americans can't distinguish between Chinese, Japanese and South Koreans."
"No matter how yellow our hair is dyed or how sharp we change our nose, we can't become Westerners. We should know where our roots are," Wang told the main group of guests, according to Chinese daily Global Times.
AFAIK, North Korean or Russian actions are never painted in those "Cold War mentality" terms though, by Chinese officials.
OTOH, Xi Jinping has rather transparently denounced the US and its allies for imposing sanctions on Russia, after Russia's invasion of Ukraine, in those terms.
Nations need to “reject the Cold War mentality and bloc confrontation, oppose unilateral sanctions and abuse of sanctions, and reject the small circles built around hegemonism by forming one big family belonging to a community with a shared future for humanity”, Xi was quoted as saying by the official Xinhua news agency. [...]
Xi said imposing sanctions could act as a “boomerang” and a “double-edged sword,” and the global community would suffer from “politicising, mechanising and weaponising” global economic trends and financial flows.
Yeah, if you really squint there, the first part of that phrase might be a veiled swipe at Russia too, but it's a far more obvious one against NATO.
In somewhat less official/high-profile sources [but probably officially sanctioned to some extent], Chinese media explains that not condemning Russia's invasion is part of the core tenets of China being in the non-aligned movement.
Non-alignment is both the core and a key premise, demonstrating China's independence in making policies and refusing to attach to or submit itself to others. China does not ally itself with any country, including Russia.
The principle of non-alignment, non-confrontation and non-targeting of any third party is enshrined in the Treaty of Good-Neighborliness and Friendly Cooperation Between China and Russia signed in 2001.
(N.B. China is officially allied with North Korea in a formal military sense of a having a mutual defense pact. I'm not sure why China Daily glosses over that.)
OTOH economic globalization is described in materialist-dialectic terms as the inevitable future of mankind (both by Xi and some Chinese premier), so anyone opposed to China's vision of globalization has cold war mentality (according to them).
“Economic globalisation is an objective requirement for the development of productive forces and an irresistible historical trend,” Xi said.
or
“We have to abandon the Cold War mentality, try to understand the essence of things from the perspective of material duality, endeavor to build a community with a shared future for mankind, and join hands to respond to global challenges,” Liu said, according to a translation. “We believe that an equitable international economic order must be preserved by all of us.”