Approximation factor shorthand clarification
That article, Linear-Time Approximation for Maximum Weight Matching calls "a matching $\delta$-approximate, where $\delta \in [0, 1]$, if its weight is at least a factor $\delta$ of the optimum matching."

That $\delta$-approximate maximum weight is, in fact, similar to the 2-factor approximation that is easy for you to grasp. When the optimization problem is to minimize some (positive) quantity, such as the minimum traveling distance or minimum vertex cover, then a 2-factor approximate might appear. When the optimization problem is to maximize some (positive) quantity, such as the case here, then a $(1-\frac15)$-factor approximate might be expected.

You may want to take a look at the performance guarantees of approximation algorithm or even what is a bicriteria approximation algorithm?