Bain doesn't have a global test, so this will vary by geography. Different offices have also been trialling different types of test recently so it's really hard to say. Generally though they are standard "quantitative" tests that are either similar to the GMAT or to a standard SHL test - https://www.shldirect.com/en/practice-tests. The test could also have a verbal reasoning section.

BCG also has different test types. In London they have a standard "quantitative reasoning test" wihch is very similar to GMAT Problem Solving questions. They also have a "BCG Potential Test" which is much more of a "full length case" with a number of questions to answer. Some will be quantitative, others will be about applying judgement, critical reasoning and reading comprehension. In this sense it is much more similar to a McKinsey PST (although the PST is still adminstered on paper while the Potential Test is an online test.

Hope that was somewhat helpful!