While, as an adult, serving as the commanding officer of the NX-class Enterprise in 2153, Archer supervised a boyhood Sim – essentially a clone of Enterprise's chief engineer, Charles Tucker III – while the latter remotely flew the same model in Enterprise's launch bay, during which the craft was also closely watched by the captain's dog, Porthos. Sim found that the model kept pitching slightly to starboard and, though he complied with advice from Archer that he should trim the craft's port aileron, doing so did not stop Sim from accidentally crashing the miniature, just as Archer had done repeatedly before him. This collision caused the breakage of the model's starboard nacelle, but Archer assured Sim that the damage was "nothing a little glue won't remedy." Sim was indeed extremely eager to fix the craft, even moments after learning the eventful news that he was a mimetic simbiot. (ENT: "Similitude")
Jonathan Archer's model spaceship was one of few ship designs that only showed up as a model and not as an in-universe ship. The script of "Broken Bow" characterized the model as "a scale model of an early 22nd century Starfleet transport." The vessel's name and number were scripted to have been painted on the model's hull.  However, this is not the case in the final version of the episode.
The remote-controlled toy was one of several props from "Broken Bow" for which illustrators John Eaves and Jim Martin were instructed to create alternative designs. Eaves illustrated a rough concept sketch of the toy as well as a slightly more detailed diagram of the model (along with a concept image of its small zero gravity device), but it was Martin who was ultimately responsible for designing the model.  He based it on the design of the Lockheed Martin X-33. ("Broken Bow" text commentary, ENT Season 1 DVD/Blu-ray)
The model was built by the Paramount Pictures prop shop. ("Broken Bow" text commentary, ENT Season 1 DVD/Blu-ray) It measured 27x33x7 inches and was custom-made out of plastic.  Some shots of Archer's model in "Broken Bow" were visualized with a CGI version of the starship, created by Eden FX. The digital replication had to match its real-life counterpart exactly but it also had several advantages, the most significant of which was that it could fly more convincingly than if the physical model had been suspended from a wire. (Star Trek: The Magazine Volume 2, Issue 10, pp. 28 & 29) Brent Kling was one of the modelers who worked on it. 
In the final draft script of "Similitude", the model was initally referred to as "starship transport model (the one young Archer flew with his father in 'Broken Bow')." However, the model was mostly called simply "starship model" in that script.
While disassembled, the model was to have been featured in a scene extension written into the "Similitude" script but not included in the final edit of the installment. In the scripted scene extension, parts of the model were shown by Archer to Sim, in a metal chest which was full of various mementos and which Archer pulled out from under his own bed. Just before opening the chest, Archer explained he had flown the model when he had been the same age as Sim was. Sim was wowed by the model, gingerly picking up one of the pieces. Archer commented, "It never held warp five, but it held its own." Sim was curious as to whether the model could still fly, so Archer suggested they find out. In the next scene (included in the episode), the model was spoken about in slightly more scripted dialogue that didn't make it to the screen. Archer admitted that he had been unsure as to whether the model would fly and congratulated Sim for having reassembled the model's anti-grav motor, but Sim remarked that doing so had been "easy."
Manny Coto, writer of "Similitude", was extremely pleased that the model reappeared in that episode, hearkening back to the pilot episode, "Broken Bow". Coto also liked the parallel between a boyhood Archer flying the model in the earlier installment and a similarly-aged Sim flying the same model in "Similitude". ("Similitude" audio commentary, ENT Season 3 DVD)
The practical model of Archer's toy starship showed up in the 40 Years of Star Trek: The Collection auction as Lot #167, described as "JONATHAN ARCHER'S MODEL SPACESHIP". Although it was estimated between US$500 and US$700, it was sold on 5 October 2006 for US$2,200 (US$2,640 including buyer's premium).