There is 1 big resemblance between the way DASH and the way MONERO provide anonymity.  The anonymity in both cases comes from the fact that a genuine transaction to an output A could potentially come from different inputs B, C or D.  This {B, C or D} is what is called the anonymity set (I take 3 elements as an example, it is not 3).  This is different in bitcoin, where an output A comes from an input B.  If you know A, then you know that B paid A.  In monero or in DASH, if you know A, you know that ONE of B, C OR D paid A but you don't know which one.

There are 3 main differences:

1) the anonymity in MONERO is mandatory and automatic, while the anonymity in DASH is an active option you have to take.

2) the anonymity set in DASH are *really spend transactions*, while the anonymity set in Monero consists of *potential* transactions (of which only one is truly happening).

3) the anonymity set in monero is generated by the transmitters' wallet on his own, while th anonymity set in DASH is generated by a trusted party (a masternode) who knows the participants.