It depends what factors you consider important/definitory for a cold war. Since (2022-23) hindsight is overbearing, just sticking with how things were in 2018:
Russia had basically won a proxy [but hot] war against the US in Syria. Granted the US wasn't terribly convinced whom to help there but it did send some weapons, so it was technically involved in that proxy war with Russia's backed Assad, albeit the US was less involved than Putin was. (Putin seemed cognizant of that, and was proclaiming the end of the "US empire" that year, by the way.)
There was another proxy war in eastern Ukraine too, started in 2014. By 2018, the US was announcing they were preparing to send weapons (Javelins in particular) to Ukraine, although those weapons had been earmarked to be used only in a case of a more serious Russian incursion (which of course, in hindsight, we know eventually happened some 4 years later.)
Even without an ideological divide of the magnitude from Soviet times, must Russians continued to see US (and NATO) as their main enemy:
"The independent Levada Center pollster published the results of its September survey on Russia’s top enemies. Some 70 percent of respondents pointed to the United States, a similar result as that in a November 2018 survey."
If you want to slice ideology, hardcore communism was never too popular in the US, but Putin's rightist anti-liberal ideology surely is more popular. So that's a difference from the Soviet times. In 2019 Putin would declare the death of liberalism, although he made similar statements before. One interesting issue that I remarked before is that although some rather illiberal parties/governments in Europe, like Hungary's Oban side with Putin on many things, others like Poland's PiS are much less inclined to do so. (The same is true in Western Europe, e.g. comparing Vox with the AfD.) Mere ideological agreement with Putin on anti-liberalism isn't a terribly reliable predictor on their stance to Russia-related matters. On the other hand, even during Soviet times, there were Chinese-aligned movements (particularly in Asia and Africa) who were at odds with the Soviet-aligned ones, even warring each other, despite broad ideological similarities.
Getting into the minutia of Trump's statements on Russia is probably rather hopeless given their vagaries over time, but some are interesting, like this one in July 2018:
“What good is NATO if Germany is paying Russia billions of dollars for gas and energy?” Trump tweeted.
Of course, I'm being rather selective with Trump's statements, but that one does highlight a difference from the Soviet era, i.e. much broader economic engagement of Europe with Russia, and the American displeasure with that, which wasn't limited to Trump's presidency, by the way.