"Cover" has several meanings. This is definitely a pun ("joke or type of wordplay in which similar senses or sounds of two words or phrases, or different senses of the same word, are deliberately confused") as the phrase is printed on an umbrella, and an umbrella covers; and presumably the supplier of the umbrella provides some kind of coverage too.
Certainly, one of the definitions of "cover" can apply to an umbrella, but there are several other meanings as well:
"Cover" could be used in the same way as it is during military operations, where a single soldier, or group of soldiers venture into the open, while another group of soldiers "cover" them from behind, that is, provide gunfire to distract the enemy and protect them.
"Cover" could also mean "provided for", e.g. "The contract covers all the costs..."
In this case, of the umbrella, the meaning could be interpreted differently as the individual pleases. But the main point of the humour is that the words have more than one meaning, and that all their meanings are valid the way the sentence is phrased.