As far as I understand, in Gintama, aliens invaded Edo (Tokyo) in the times of the samurais and defeated them. Then, we not only see samurai-era (Edo period) buildings, weapons, etc., but also alien-made buildings, weapons, and technology.
However, there is also technology from the current era in it, like cars, TVs, electricity cables, bazookas, motorcycles, etc., like if the current era would be also mixed with their time.
How come there is contemporary technology in Gintama universe?
While it is true that aliens did conquer edo and enforced their rules and regulations, the bakufu reserved the right to rule their subjects while conforming to the alien government. By preserving the current era's tech they may try to make it seem like the humans of current edo may have some control over their land.
It can also seem like a form of resistance. This can be seen in the episode where the Dango shopkeeper hasn't closed his shop despite being the only one in the vicinity and all the other shops have either adapted the new alien sweet technology or have closed and moved on.
The other explanation I could think of is that they wanted create a story line that wasn't too complex and kept the series lighthearted like with the episode where Gintoki wants to buy an old fan because he is too cheap to actually invest in an AC. All of this is just my interpretation, I'm not sure if anything about this is actually specified in the manga.
I do believe it's something akin to a mix of both worlds - Alien futuristic technology might have accelerated the development of cars and other modern technology. It's how I interpret it, anyway.
