Let's start with the year TWEWY's events happen. TWEWY on the DS has no indicator in the game but I think it is fair game to assume it takes place in the year it was released. We can, of course, see technology of that era (FLIP PHONES!!!).
Years later the animation happened and things changed. The animation was modernised as we can see from the updated mobile phones now being smart phones. The anime takes place not in 2007 but in 2021. The creative team even stated that the sequel of TWEWY (NEO TWEWY) is a sequel to the animation and not the OG release, most likely to explain the technology leap and also changes to Shibuya.
Now for the month of OG TWEWY. It is something I speculated about with other TWEWY fans. While playing I entered a battle and a sign was shown with an October 8th date. Great! So, TWEWY takes place in October, right? Then I went into a search for dates shown anywhere in the game and I found some more. There are two signs in West Exit Bus Terminal that have March 4th - April 5th & February 3rd - March 4th dates. That... confused things. Later, I found a poster at Udagawa Back Streets with a March 21st date. So, yeah, that didn’t help to reach a result. After that I found that there are characters thinking about Christmas. March is so early to be thinking about Christmas so... my (surely not very accurate but fun to speculate) theory is that the October sign is a leftover sign, TWEWY’s timeline is nearing Christmas (maybe November / early December) and there are promotion signs and posters that promote events coming in the first months of the new year. Let’s not forget you can buy a New Year’s Gift at the shop.
*The above info concern the localisation and not the Japanese release.