Does trimmed SIM card affect iPhone 4S's signal strength?
No - the sim card only provides an ICCID number to the phone. That number is read once per boot as the phone just uses the sim to identify what account to bill for service when registering with the nearby cell towers (IMEI and ICCID together let the system know what number to assign your phone and who to bill). 

Something else must be causing your observed behavior. 

Keep in mind, there is no guarantee two phones in the same location are not on distinct cell towers. Especially if one tower is overloaded or doesn't have high speed data so the phone /carrier might pick a faster connection that happens to have less bars of signal. 

What does your carrier say if you call them for a site survey / diagnostic?