The EU has a harmonized CPI methodology (HICP). It has sometimes been applied (e.g. in OECD/G-20 contexts) to the US for comparison purposes. There's more to methodological harmonization than what goes in the basket (e.g. which areas are covered or the formulas used for aggregation).
There's also an attempt to standardized the basket in UN's COICOP; alas few countries use that as-is.
Some selected quotes from a G-20 CPI paper, which is quite detailed:
The preference for the use of the HICP, when available, rather than the national CPI is
founded on the fact that Eurostat has developed a harmonised approach and a single set of definitions
in order to arrive at comparable measures of inflation in the EU. Given the higher comparability of the
HICPs, the use of HICPs when available is considered appropriate for the compilation of the CPI G-20
aggregate. The availability among G-20 economies is as follows:
- HICPs for the European Union covering all current 28 Member States: data from January
1996 (including HICPs for France, Germany, Italy, and the United Kingdom);
- HICP for Turkey: data from January 1996;
- Experimental HICP for the United States that differs from the US National CPI in two major
aspects. The US HICP (i) includes the rural population, and (ii) excludes owner occupied
housing. However, for reasons of prominence, main uses and timeliness (the US HICP is
published with a delay of one or two months with historical data starting in January 1998
only) the use of the US national CPI is preferred in the calculation of the G-20 aggregate.
France, Germany, Italy, and Turkey publish both HICPs and national CPIs.
methodology used for the HICPs differs from the methodology used for the compilation of the
national CPIs. [...] In general terms, the main methodological differences between the HICP and the national
CPIs are as follows:
1. Coverage: Excluded from HICPs are: Narcotics (COICOP / HICP 02.3); Imputed rentals for
housing, including rentals of owner-occupiers (04.2); Games of chance (gambling) (09.4.3);
FISIM (financial intermediation services indirectly measured) (12.5.2); and part of personal
care services n.e.c. (12.1.3 ) (example: prostitution);
2. Treatment of subsidized healthcare and education: the HICP includes the net price paid by
consumers (after reimbursements);
3. Formula used for elementary aggregates: the HICP use is ratios of arithmetic mean prices
and/or ratios of geometric means;
4. Geographical and population coverage: the HICP covers all expenditures on the territory of
the country concerned, whether made by residents or non-residents (e.g. tourists).
Regarding baskets in (other) G-20 countries:
The COICOP classification of consumer expenditures and prices on goods and services by
purpose is only strictly applied by Korea and South Africa, while the COICOP/HICP classification
(which is the COICOP classification adapted to the needs of the HICP) is used for the European Union
and by Turkey. The other G-20 economies use national classifications which in some countries, such
as Australia and Japan are consistent with COICOP at the highest level. Table 7 gives an overview of
the systems of classification used and highlights the specific exclusions or inclusions by country.
There is more to say about housing costs as well, in terms of methodological variation between countries, but I'm omitting that here; the paper covers it in detail too.
Also the CPI aggregation method may differ between countries:
The compilation of a CPI consists of two main stages: 1) the calculation of unweighted price
indices (mostly price ratios) for elementary aggregates, and 2) the weighted aggregation of these
elementary price indices for higher level aggregates. [...]
Methods used for combining prices to give elementary
aggregates vary widely among the individual countries of the G-20, the geometric mean (Jevons
index) is the method most commonly used (10 countries plus France, Italy and the United Kingdom).
All G-20 economies use a Laspeyres-type formula for their published CPIs [but there are some variations in details for that as well].
There's also a table with those aggregation/index formulas per country, in the G-20 paper.