We investigate the properties of the mixed-mode (RRd) RR Lyrae (RRL) variables in the Fornax dwarf spheroidal (dSph) galaxy by using B- and V-band time series collected over 24 yr. We compare the properties of the RRds in Fornax with those in the Magellanic Clouds and in nearby dSphs, with special focus on Sculptor. We found that the ratio of RRds over the total number of RRLs decreases with metallicity. Typically, dSphs have very few RRds with 0.49≲ P0 ≲0.53 d, but Fornax fills this period gap in the Petersen diagram (ratio between first overtone over fundamental period versus fundamental period). We also found that the distribution in the Petersen diagram of Fornax RRds is similar to Small Magellanic Cloud (SMC) RRds, thus suggesting that their old stars have a similar metallicity distribution. We introduce the Period-Amplitude RatioS diagram, a new pulsation diagnostics independent of distance and reddening. We found that Large Magellanic Cloud (LMC) RRds in this plane are distributed along a short- and a long-period sequence that we identified as the metal-rich and the metal-poor component. These two groups are also clearly separated in the Petersen and Bailey (luminosity amplitude versus logarithmic period) diagrams. This circumstantial evidence indicates that the two groups have different evolutionary properties. All the pulsation diagnostics adopted in this investigation suggest that old stellar populations in Fornax and Sculptor dSphs underwent different chemical enrichment histories. Fornax RRds are similar to SMC RRds, while Sculptor RRds are more similar to the metal-rich component of the LMC RRds.