The methylenedioxybenzene ring system is found in many natural products (such safrole, used by the Cherokee as a treatment for syphilis) and medicines (such as the antibiotic Ciproflaxin) because of chemical stability (aromaticity) and easy of biosynthesis (from the common amino acid tyrosine). It is used recreationally to get high in MDMA because of its resemblence to DA and its interaction with DA's, which it physically resembles, receptors.
Huckel's rule of aromaticity (unusually large stability) of an organic chemical compound is defined as a compound that is flat, planar, circular (for n>0) and one that has H(n)=4n +2 pi electrons. If n=0, then H(0)=2 and we have the unusually stable/hard to separate case of the lone pair of electrons. If n=1, then H(1)=4(1)+2=6 and we have examples such as thiophene, pyrrole, furan, and benzene. If n=2, then H(2)=4(2) + 2=10 and we have azulene and 1,3-benzodioxole (the IUPAC word for 3,4-methylenedioxybenzene).
The biggest catch here is that the two H's of the methylene unit are basically at 45 degree angles to the plane of the molecule at large (one up and one down; this is of minor importance) and that the C of the methylene unit is not a conductor of electrons but rather an insulator of electricity. Still, this addition of a CH2 to a catechol (a vicinal di-hydroxy aromatic chemical functionality) makes the two O's more in the plane and allows one each of their two per O lone pairs of electrons to better geometrically and electronically line up with the circuit of the parent benzene ring. Also, being connected as two rings within a larger ring allows for better vibrational communication leading to, in some cases, greater resonance energy advantages.
From a biological/biochemical/evolutionary perspective, dopamine is synthesized from tyrosine and tyrosine is one of the naturally occurring, ubitquitous 20 amiino acids found after the break down of any protein, such as meat. The methylene unit added to the dopamine derived from tyrosine (each with its generally somewhere to be found -CH2CH2NH2 side chain harking back to its days as a subunit of a protein) also makes the 1,3BNZDXL compound less polar and therefore better able to cross lipid filled membranes such as that of spirochetes or of the mammalian blood brain barrier, helping both sarole, Ciproflaxin and MDMA do their jobs better.
The 1,3-BZDXL/3,4MDOB bicyclic ring system is everywhere because it is both stable/easily formed AND useful.