C182 pilot reports entering the runway and taking off without clearance after misunderstanding an instruction to an aircraft on landing roll to clear without delay for traffic on final. The reporter's aircraft and the aircraft on rollout had similar sounding call signs. The aircraft on final is instructed to go around.

Narrative

In the run-up area waiting for departure; a single engine aircraft landed while I was monitoring the Tower frequency for departure. I then heard from ATC what I thought was my call sign and a clearance for a 'no-delay' departure with an aircraft on [a] X-mile final (I cannot remember the exact distance stated). I repeated back my 'no delay' departure acceptance with approaching traffic in sight. Crossed the hold-short threshold and immediately departed. Shortly after rotation I heard the approaching traffic announce they saw traffic on the runway and ATC issued a go-around. I then continued on my planned flight with flight following. Upon return; I was asked to contact the Tower from Ground Control while returning to the ramp. Subsequent communication with the Tower indicated that the call sign (albeit ending similar) was directed at the aircraft exiting the runway; and that my call sign was not given a departure clearance. I believe the possible deviation was caused by the fact that I was the only aircraft in the run-up area and the call sign I heard 'sounded' like my call sign (thus I perceived it as such); along with the fact that I have on multiple occasions been given a 'no-delay' departure clearance with 'traffic on X-mile final.' Furthermore; ATC did not challenge or question my read-back of the departure clearance. Nothing seemed odd or out of the ordinary for the clearance; and I was surprised I was contacted later for a possible deviation. Although I was not distracted while monitoring the Tower frequency; future prevention will be to maintain a sterile cockpit environment while in the run-up area and if there is any doubt as to the clearance; I will ask for ATC to repeat the clearance before any action is taken. In a future; similar; situation I intend to decline the no-delay and allow approaching traffic to land if I am not 100% sure of the 2-way communication.