Are there any good modern analyses capturing the multidimensional
nature of the political positions of parties and voters?
The best modern analysis of the political positions of elected officials in the United States at the federal level (from which the political positions of parties can be inferred from the political positions of their members) is the one created by a collaboration of political scientists who share their findings on an ongoing basis at Voteview.com.
Voteview.com has a data driven method of evaluating the political position of elected officials at the federal level in the United States that relies only minimally upon subjective evaluations of political issues.
Analysis of their database has revealed that for most of U.S. political history, political ideology is quite accurately captured by a simple one dimensional liberal-conservative dimension, and that for some periods of U.S. political history two dimensions of political ideology are necessary to accurately summarize the detailed floor vote by floor vote data. As it explains (with links to further discussion at the the source):
Ideological positions are calculated using the DW-NOMINATE (Dynamic
Weighted NOMINAl Three-step Estimation). This procedure was developed
by Poole and Rosenthal in the 1980s and is a "scaling procedure",
representing legislators on a spatial map. In this sense, a spatial
map is much like a road map--the closeness of two legislators on the
map shows how similar their voting records are. Using this measure of
distance, DW-NOMINATE is able to recover the "dimensions" that inform
congressional voting behavior.
The primary dimension through most of American history has been "liberal" vs. "conservative" (also referred to as "left" vs. "right").
A second dimension picks up differences within the major political
parties over slavery, currency, nativism, civil rights, and lifestyle
issues during periods of American history.
The technical details of the DW-NOMINATE model can be found in Poole's
Spatial Models of Parliamentary Voting. Poole and Rosenthal's Ideology
and Congress explores the nature of voting in Congress and the
political history of the United States through the lens of the
ideological dimensions recovered by DW-NOMINATE and is a useful
companion to this site.
For example, in the 1980s, the U.S. federal government had a de facto three party system in its legislation comprised of Northern Democrats, Southern Democrats, and Republicans differentiated in two political dimensions that has since then, in a process known as "realignment" become a more pure two party system with a single dimension of political ideology.
The same methodology has been applied by some members of that collaboration to the decisions of U.S. federal judges to rate the ideology of the judges and the average ideology of the intermediate level federal courts of appeal in the United States.
To the best of my knowledge, little work has been done to replicate this approach at the state and local level in the United States, although it would be straight forward to do so.
Likewise, the methodology itself which isn't inherently tied to a two party system and indeed is designed to highlight intra-party factions, could easily be extrapolated to more than two party multiparty parliamentary democracies. But, it would be less useful in that context because in parliamentary systems party discipline is greater which makes floor votes on bills in parliament a less useful gauge of partisan ideology.
This methodology is not well suited to evaluating the political ideology of voters since it is based upon floor votes in legislatures or judicial decision rendered in court cases.
There have, however, been numerous efforts using survey data and cluster analysis techniques to describe the political ideologies of U.S. voters in a multidimensional fashion to goes beyond partisan identification and highlights factions within U.S. political parties. The Pew Research center is among the organizations that have done such work. See, e.g., Pew Research Center, "Beyond Red vs. Blue: The Political Typology" (May 4, 2011) (identifying nine ideological clusters among U.S. voters based upon a multidimensional analysis of political survey responses summarized in the chart below).
Both Votevbiew.com and the Pew Research Center are notable for allowing the dimensions of political ideology to emerge from the data based upon the correlations between people's views on different specific issues, rather than prejudging what the dimensions of ideology upon which people differ in their views are and then trying to fit people's views into the data by subjectively evaluating the ideological status of particular issue positions.
In contrast, the Political Compass has established two predetermined dimensions (which are shown in the Financial Times illustration in the question) in advance and then trying to fit particular politicians, parties, or clusters of voter opinions onto that pre-established template. The methodology was applied, for example, to candidates in the 2020 Presidential election in the United States as follows:
A different political compass effort to place the U.K.'s political parties than the one in the question is this one:
Deciding where to put the "zero" point of true moderation, and how to scale the candidates on each of the dimensions, however, is largely a subjective matter. Also, a single dimension from the top right to the bottom left, however, captures most of the variation, since positions on both dimensions are strongly correlated in the current political environment of the United States.