Do you think that the terms "closest evolutionary relationship" and "most common ancestor" used when describing the relationship between two different organisms mean the same thing?

Yes, the above terms will be much similar.

Explanation:

The term 'closest evolutionary relationship' reflects the organism or species which were the closest to different species in a phylogeny.

Most common ancestor also describes the ancestor from which two different organisms might have evolved.

Both the terms 'closest evolutionary relationship' and 'most common ancestor' show the common ancestors in a phylogenetic lineage. Hence, on evolutionary levels, both these terms have similar meanings and are often used to describe phylogenetic lineages.