Yes. The first generation Stratus receives traffic on the 978 MHz (UAT) frequency and the second generation receives it on both the 978 MHz and 1090 MHz (ES) frequencies.
The traffic you see on ForeFlight Mobile is ADS-B traffic. ADS-B traffic only includes the air-to-air traffic received directly from another aircraft and traffic rebroadcast to your aircraft from an ADS-B ground tower. Air-to-air traffic reports are only reported from ADS-B Out equipped aircraft. ADS-B Out must be installed in your aircraft, and cannot be provided by a portable device like Stratus. Few aircraft are currently ADS-B Out equipped.
You will only receive ADS-B traffic from a ground tower if two things are true:
1) your aircraft, or another one nearby you, is ADS-B Out equipped, and
2) you are in communication with an ADS-B ground tower.
Remember, ADS-B traffic, even at its best, is not a complete picture of nearby traffic and is not a substitute for looking out the window when fulfilling your responsibility to see and avoid.
Video 1: ForeFlight and ADS-B Traffic Part 1.
Video 2: ForeFlight and ADS-B Traffic Part 2.