It's a somewhat complex topic (hence the belated answer). Ideological polarization between political parties has been measured for years (in various ways). From a recent paper summarizing the approaches:
Definitions of political polarization usually involve distances in policy stances or attitudes between groups of people, parties, or specific representatives. In his classic study, Giovanni Sartori (1976) identified political polarization as the ideological distance between candidates, parties, and/or voters. [...]
Many studies have relied on weighted variance calculations to devise comparative indices of political party system polarization (PSP; Dalton, 2008; Ezrow, 2007; Hazan, 1995; Pardos-Prado & Dinas, 2010; Taylor & Herman, 1971). A frequently cited formula in recent years was developed by Russell Dalton, using data from the Comparative Study of Electoral Systems, focusing on ideological distances between parties in legislatures as perceived by the public. This formula demonstrates the distance of mean party positions on the Left–Right scale from the mean ideological (Left–Right) position of the legislature as a whole (Dalton, 2008). [...] Ladner (2014) takes this method one step further, measuring spread of party positions in relation to average party system positions, but over multiple issues, and then averaging the scores of all dimensions to produce an index of political system polarization by country-year.
For a more in-depth discussion of measures in the style of Dalton, see Schmitt (2016); there's a giant figure in there giving a visual comparison of the correlations between various indices (all calculated using the same underlying dimensions & data).
These measures generally assume an expert-style rating of parties position on several dimensions. Ladner uses seven dimensions for example: economic liberalization, financial policy [I guess he means fiscal], law and order, immigration policy, environmental protection, welfare state, and finally "liberal society". The ratings can be given by experts or based on public perception. The choice of dimensions too can (of course) produce different results.
Some scholars, for example, have found that voters have difficulty correctly locating parties on issue scales (Carpini & Keeter, 1993; Iyengar, Sood, & Lelkes, 2012). Zechmeister (2015) found that in Latin America, many voters either cannot or will not place themselves on a Left–Right scale [...]
Also, the polarization thus measured between political parties and the public at large has shown little correlation (at least in some studies):
Political elite polarization, however, is not undeniably linked with polarization among the public (Abramowitz, 2010; Fiorina, Abrams, & Pope, 2005). [...] political party polarization measure does not necessarily fully capture polarization at the mass level of society. Pardos-Prado and Dinas (2010), for instance, measured both the party system and mass ideological polarization of European countries by variance of ideological positions, finding the correlation between party system and mass ideological polarization to be relatively weak at r = 0.3.
Another approach to measuring party polarization draws on legislative voting rather than mass public perceptions of party ideological positions. Scholars studying the United States have tended to follow this pattern, creating Dynamic Weighted NOMINAL Three-step Estimation (DW-Nominate; Poole & Rosenthal, 1997) to measure Congressional roll-call votes as indicators of within-party ideological homogeneity and between-party ideological distances. More than two decades of research has shown that the ideological distance between parties in U.S. legislatures at the state and national levels has been growing over time (Davis & Dunaway, 2016; McCarty, Poole, & Rosenthal, 2006; Poole & Rosenthal, 1997; Shor & McCarty, 2011).
The upside of NOMINATE-style methods is that they don't presuppose a number of dimensions (these are extracted from the voting data). The downside is that NOMINATE-style methods are not readily applicable to the general public as there's no meaningful notion of "legislative record" for individuals in the public at large.
As a sort of a compromise, the paper I've been quoting from (Lauka, McCoy and Firat, 2018) proposes a measure of public polarization that
focuses on polarization over parties rather than ideology or any other single political issue [and thus] fits well with the multidimensional nature of political polarization and represents ideological as well as nonideological divisions within societies.
As this is a new method it hasn't been widely used. I see a potentially the issue that the electoral systems of countries, which are generally assumed to affect the number of parties (see Duverger's law) confounding some dimensions. The paper doesn't mention/discuss the issue, as far as I can tell.
As was hinted in the opening quote, another way to look at polarization is the affective dimension. This pretty much treats politics like preferences for (say) football teams, i.e. without trying to get to any ideological roots. At least regarding US politics, this approach has been increasingly popular. See Iyengar et al. (2019) for a recent review. For simpler explanation using "thermometer" style measures, see this Pew report and even maps of that in The Atlantic. As far as I can tell these (affective-focused) methods have not been widely used to study European politics, but there are some recent exceptions with intriguing findings:
The aim of this article is to study the concept of affective polarisation in European party systems. It introduces the Affective Polarisation Index (API) that allows for measuring and comparing levels of affective polarisation also in multiparty systems. This novel measure is applied to 22 European democracies and the United States between 2005 and 2016. The results indicate that affective polarisation is acutely present in European party systems, as partisans are often extremely hostile towards competing parties. The most affectively polarised countries are in Central Eastern and Southern Europe where the degree of affective polarisation is notably higher than it is in the United States, while Northwestern European countries are more moderate in terms of partisan feelings. Further analysis reveals that affective polarisation is significantly correlated with ideological polarisation, but the relationship between the two appears to be conditional: in some Western European political systems ideological polarisation does not lead itself to strong interparty hostility, while in Central Eastern Europe a high degree of affective polarisation can be present even in ideologically centrist party structures. These findings validate the claim that ideological and affective polarisation are two distinct aspects of polarisation, and that the latter also merits additional attention.
Measuring affective polarization in a multi-party (>2) context seems to produce quite different results depending on the metrics used though. Unlike the above (Reiljan, 2019), which used an all-to-all metric, Gidron, Adams, and Horne (2018) using only a left-right metric conclude that even in Western Europe affective polarization is higher than in the US...
in several countries including Spain, France,
the UK, and Switzerland, supporters of the largest left- and right-wing parties expressed more
intense mutual dislike in every CSES election
survey we analyzed than did the American
Republican and Democratic supporters in any
of the CSES surveys
This is actually somewhat consistent with an empirical observation of Harteveld (2019) that on an all-to-all measure (like Reiljan's) more parties in a country/system correlate with a lower affective polarization. The intuitive explanation of that is that
self-identification with a single party will often be
smaller if multiple parties exists that are ideologically close. As a result, a partisan ingroup is not
opposed to every possible partisan outgroup to the same extent. A green voter does probably not
perceive a typical social democratic voter as a disliked outgroup, or at least not to the same extent
as (s)he might consider a conservative voter – let alone a populist radical right voter
Although most affective polarization studies use "thermometer" style scales in their raw data, some have used [trust] game-based measures. One 2018 paper (Westwood et al.) that did that in a cross-country setting found not much difference in four Western countries across party lines, but did find differences across other lines (a nice thing about affective polarization is that it can also be used to compare other divisions in society):
Although in Belgium and Spain measures of divisions due to partisanship overlap those due to social divides, we do not find that partisan animosity is larger in these cases [...]. Also, contrary to expectations, the scope of the partisan divide is similar across the four nations [the other two being US and UK], with all at nearly one unit of currency.