<Does "congeneric" mean with identical DNA?>

No, the root here is "generic", not "genetic".  Congeneric means "belonging
to the same genus".  If two species in a given genus show marked
differences, taxonomists might question whether they are truly congeneric,
or whether they should be assigned to separate genera.  The same question
might be asked of  two very similar species currently assigned to separate
genera.  Should the two be so classified, or are  they actually  congeneric?
Similarly, conspecific means "belonging to the same species".  One might ask
whether two very similar forms are conspecific.  In other words, should they
be classified as two distinct species, or considered forms of one species
(conspecific)?
Obviously, any two forms or individuals that are conspecific are also
congeneric.

Paul M.