It was surprisingly hard to find sources that somehow relate the political positions of US candidates to the European system—in most cases it wasn’t clear whether a left/right distinction was based on the European or the US perception. Nonetheless, here are two examples:
Bernie Sanders describes himself as a ‘democratic socialist’ and is perceived to be quite close to the leftmost end of the US spectrum. In 2016, while he was competing in the Democratic primaries against Hillary Clinton, the German Press Agency (DPA) described him as:
In Deutschland wäre Bernie Sanders mit seinen Ansichten wohl Sozialdemokrat.
(Translation: in Germany, Bernie Sanders would probably qualify as a social democrat.)
On a left-right spectrum, the social democratic SPD used to be the large left-of-the-centre party with both more leftist and more liberal* wings. The classic socialist and communist parties have always been firmly on the SPD’s left as is the Left Party, the successor of the former state party of the socialist GDR. The authors did not state which wing of the SPD they would put Sanders into but my personal assumption would be that he would neither be in the left nor in the liberal wings of the party.
Joe Biden has been described by Donald Trump as a Trojan Horse for socialism. The (editorially conservative) magazine Focus used that quote to introduce an article on the election’s likely effects on the stock market dating from 21/10/2020. It continues:
Joe Biden kontert: „Sehe ich aus wie ein Sozialist?“ Wohl kaum. Gemessen an europäischen Maßstäben, ist Biden allenfalls ein sehr gemäßigter Sozialdemokrat.
(Translation: Jo Biden counters: ‘Do I look like a socialist?’ Not at all. According to European standards, Biden would be a very moderate social democrat at best.)
The way the author phrases this allows Biden to just end up on the very liberal side of the SPD; most likely he would feel more at home in a more liberal party such as the FDP. Thus, we would have somebody who stands rather close to the political centre; he might be just left but he’s slightly more likely to be just right of the centre.
It’s worth pointing out that in neither of the two are classified as conservative. That agrees with my interpretation: politicians in the Democratic Party are unlikely to be conservative (but there may be some on the party’s right wing fringe that could classify as conservative). In general, they are varying degrees of liberal, sometimes social-liberal or in extreme cases social democratic. On the other hand, the Republicans in general tend to be seen as more conservative than the main conservative party, the CDU—but again, some Republicans trend well into the liberal spectrum while others are clearly even further right so that they would have left the area called conservative.
Why do we see this that way? There are a number of features especially in employment law that are common all across Europe which don’t exist in the US, most notably mandatory (typically government-organised) health insurance, a legal minimum of vacation days, minimum notice periods (no hire&fire), etc. These are considered achievements of the political left and the general political debate is whether to expand these rights and benefits (position of the left) or to curtail them (position of the liberals and conservatives) but there are only fringe positions that want to abolish them. As there is not even a consensus across the entire Democratic Party for ‘Medicare for all’, the party cannot really be considered left. On the other hand, as there is strict opposition to a European-style health insurance system in the Republican Party, they essentially placed the label conservative on themselves.
Obviously, there are numerous political issues across countries that do not directly touch workers’ rights and where thus the left/right divide is less clear. For example, if one considers combatting climate change the positions in German politics range from ‘a little and slowly’ (CDU, SPD) to ‘fast and now’ (Green party) which is more of a camel hump on the traditional left/right axis. In the US, for some reason this policy area has a clear ‘right end’ (don’t do anything/it’s not real) and a clear ‘left end’ (act now and fast).
Furthermore, there are social issues such as abortion or marriage equality where the parties are much more likely to move along with the general consensus in a country. Thus, the CDU is more progressive on these social issues than the entire Republican party, since Germany as a whole is more progressive than the US (Germany lacks a strong modern-day religious conservativism). On these scales, the joke you mentioned makes most sense but obviously they only make up part of the full picture.
* Outside of the US, liberal is most often understood to mean classical liberalism, i.e. something bang in the centre of the political spectrum, most concerned with economic liberty than anything else.