This paper proposes a novel dual-frequency-band millimeter-wave extended interaction klystron amplifier (EIKA). It is primarily based on the multimode operating mechanism of dual-2π mode. This design integrates a broadband traveling-standing-wave mode input cavity with a dual-2π standing-wave mode output cavity, resulting in a compact slow-wave structure design that efficiently operates within a total circuit length of approximately 24 mm. Particle-in-cell simulation results reveal that under a 15.6 kV, 1 A electron beam and a uniform 0.6 T magnetic field, the device achieves output power for 183-1024 W across a broadly 1.20 GHz bandwidth, spanning 93.76-94.96 GHz. Remarkably, it facilitates dual-band output in both lower-2π and upper-2π bands, delivering maximum gains of 37.09 dB (1024.10 W at 93.90 GHz) and 35.75 dB (752.20 W at 94.84 GHz), with -3 dB bandwidths of 0.33 GHz and 0.20 GHz, respectively. The effectiveness for the dual-2π mode design is further confirmed through a cold-test experiment using the perturbation method. This experiment demonstrated typical dual-2π mode field distribution profiles, affirming the design's efficacy.