To me compare and liken can be used interchangeably in this specific case, however in many other contexts when X is being compared to Y, the whole point of doing the comparison is to identify differences, not similarities. Additionally, I guess the verb analogize is rarely used in English (either in AE or BE,) so we can even forget about it. PS.

Understanding the Context

Can we use "compare" with more than two things? I want to compare between the shifts in style by an author in a trilogy (three novels) and how the translator reflects those shifts in his translation. Compare to: to show likeness between two unlike things. In the poem, she compares her loneliness to a ship lost at sea.

Key Insights

Compare with: to analyze two liken things. In the poem, she compares her own lonely apartment with her sister's happier home.