C-3PO is has traveled all over the galaxy in the Star Wars films, leaving the indelible impression of his fastidious and ceremonial personality with the people he crosses paths with. However, this impression was not left on Owen Lars, who did not appear to recognize C-3PO in A New Hope after having grown up with the protocol droid on his farm.
The continuity of the main saga gets a little murky with R2-D2 and C-3PO being constants in all the films. Many people are quick to pick out that Obi-Wan Kenobi didn't recognize R2 in A New Hope after having a storied past with him in the Clone Wars. C-3PO also joins the ranks of aggrieved droids, as he spent time on the Lars' family farm in Attack of the Clones but there is no recognition of him when he meets Owen years later after being sold by the Jawas back to him. While Owen's failure to recall Threepio is puzzling, there are a few reasons that serve to pardon his forgetfulness.
In Attack of the Clones, C-3PO was sold to Cliegg Lars, Owen's father, when Watto sold Anakin's mother Shmi to Lars. C-3PO worked on the Lars moisture farm even before Owen was born. Surely, the two would have been familiar with each other as the boy grew up. After C-3PO was whisked away on adventure after adventure, he returned to Tatooine in A New Hope and was sold back to a grown up Owen Lars, who shows no remembrance of him.
The actual answer as to why doesn't Owen remember C-3PO is that it's a plot-hole in the saga, since George Lucas wrote A New Hope with little to no idea of where the droid's long and winding backstory would lead him. However, there can be some convincing in-world arguments made for why there is no recognition of the unforgettably painstaking protocol droid.
First, when C-3PO worked on the Lars' farm in Attack of the Clones, he had a tarnished silver plating. In A New Hope, he is back in his usual shiny gold attire. As evidenced in the world of Star Wars, the galaxy is chock full of 3PO series protocol droids, making the odds of C-3PO being the same droid that worked on the farm decades ago incredibly slim. Owen could have easily assumed that these were two different protocol droids if he didn't put much thought into it.
While C-3PO seems to have a discernible, precise personality and distinctive voice to fans, these might be common traits among 3PO series protocol droids. To keep a moisture farm running, the Lars family likely went through dozens of droids through the years. This makes faces and time in the desert blend together, not to mention that it had been about 20 years since Owen last saw Threepio. The lack of recognition could have come from an old man simply not remembering the passing droids of his youth.
Lastly, Owen didn't seem the sentimental type to form any emotional attachments to droids. Being a practical man, he viewed them as tools to be used and replaced when they no longer functioned. He could have recognized C-3PO as a tool from his past, but showed no outward signs of remembering him because he thought it unessential. Remembering the droid would have been a purely emotion-based moment for Owen that served no purpose in his eyes.
Regardless of Owen forgetting Threepio, it doesn't take away from the parallels Lucas drew for Anakin and Luke in the saga. Both boys grew up looking at two dusty suns in the sky, lost one of their hands and have fixed up a busted protocol droid. C-3PO returning to the Lars farm decades later is just another cyclical motif Lucas created to link the rise and fall of legends across the vast galaxy.