Final answer:
The phrase in question exemplifies a metaphor, a literary device that directly compares two unlike things without using 'like' or 'as'.
Step-by-step explanation:
The phrase 'We wrapped ourselves in the cloak of his exploits' is an example of a metaphor. A metaphor is a literary device that makes a direct comparison between two unlike things by stating that one is the other, without using 'like' or 'as'. In this case, the metaphor suggests that they are enveloping themselves in his achievements, though not literally wrapping in a cloak, to perhaps feel protected or inspired by his actions. This differs from a simile, which would use 'like' or 'as' to make the comparison; hyperbole, which would exaggerate something to a significant extent; and personification, which would give human-like qualities to non-human objects or ideas.