The USA never even declared the Vietnam "conflict" as a war, or Korea, come to that, though that did at least have the backing of the UN.
As soon a country agrees to enter a conflict on a side, which the original axes declare to be a war, it's at war. You can tell the media whatever you want of course.
It's not just the US, very few wars have been formally declared after WW2, because we all learned war is bad™, so we added more and more rules (both international and national) to make it harder to do it.
But the reasons wars existed didn't go away, so this just resulted in more and more people getting killed in "special military operations" or similar things. See e.g. "Why States No Longer Declare War"[0].
[0] https://www.researchgate.net/publication/228896825_Why_State...