Lines $CF,GE$ are parallel to diagonal $DB$.

Are the cyclic quadrilaterals $CBGDC,FBGDF$ (with same sides and angles) congruent? If not, how are they be described?

• It would be clearer to spell out the four cyclic quadrilaterals you are concerned with. Clearly $CBGD$ and $FDEB$ are congruent (by reflection), but $CBGD$ and $FBGD$ need not be (assuming distinctness of $C$ and $F$ and of $G$ and $E$, as the illustration indicates). – hardmath Sep 19 '16 at 6:49
• Changed to include only two, since the others are anyhow symmetric. – Narasimham Sep 19 '16 at 7:36

Yes. Draw the line $l$ through the center $O$ perpendicular to $DB$. Then $l$ passes through the midpoint of $DB$. Moreover $l$ is orthogonal to $CF$ and $GE$ because the two latter chords are parallel to parallel to $DB$. Therefore $l$ passes through the midpoints of $CF$ and $GE$. Consequently a reflection in $l$ maps one quadrilateral into the other.
• Note that this is the symmetry I called attention to in my Comment, and which led to a clarification of the Question that obviates its consideration. The reflection pointed out does not produce congruence of $CBGDC,FBGDF$ because the "lower" triangles are not swapped. – hardmath Sep 20 '16 at 0:05