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Is there a measure of similarity for economies?

For example, I would like to find the world economies most similar to the UK.

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To the best of my knowledge, there isn't. Ant that is actually a good thing, the economic structure of a country is usually a fairly complicated thing to be able to simplify it with a single number. You could, on the other hand, create an aggregate of measurements and try to look for patterns, e.g. you can see the behavior of macroeconomic variables such as gdp, inflation rate and unemployment, and based on those decide which countries are similar to each other.

And that is why clustering analysis is for. In the plot below I used Principal Component Analysis (just to reduce the dimensionality of the problem and be able to graphically display what's going on) to show 4 clusters fo countries. Each color is a k-means cluster, the axis are just the two firs principal components $x_1$ and $x_2$

enter image description here

I also added some of the countries on each cluster so you can see their similarity. For example, the $\color{blue}{\rm blue}$ cluster has large gdp, low unemployment, whereas countries in the $\color{green}{\rm green}$ cluster are not doing so well: high inflation, high unemployment and low gdp.

The cluster that you are interested in has these values (as of 2017)

                 country     gdp  inflation  unemployment
---------------------------------------------------------
9              Australia   55692       1.95           5.3
13               Austria   47347       2.08           5.1
17               Belgium   43488       2.13           6.2
15                Canada   45094       1.60           5.6
8                Denmark   56630       1.15           4.8
14               Finland   45927       0.75           7.2
20                France   39932       1.03           8.9
16               Germany   44769       1.74           3.3
3                Iceland   70248       1.76           2.9
4                Ireland   68710       0.34           5.3
19                Israel   40272       0.24           3.7
22                 Japan   38448       0.47           2.4
0             Luxembourg  105863       1.73           5.0
11           Netherlands   48555       1.38           3.5
18           New Zealand   41572       1.85           3.9
2                 Norway   75389       1.88           4.0
5                  Qatar   61024       0.40           0.6
12            San Marino   47595       1.05           8.0
10                Sweden   52925       1.79           6.3
1            Switzerland   80637       0.53           2.4
23  United Arab Emirates   37732       1.97           1.6
21        United Kingdom   39800       2.69           4.0
6          United States   59792       2.13           3.9
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