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If this happened with one alien race vying to be like a second one, then the first would work to adapt their technology to match, or work with, the technology of the second; keep in mind, the Earth analogy involving beings from the same race, but different cultures, while with aliens there would most likely be not only different cultures, but different physiologies, and different environments. For instance, a race on a planet with strong gravity, and a much stronger magnetic field, would most probably have to make adaptations to technology from a planet with weaker gravity and a much weaker magnetic field in order for the technology to work.
Thus, if an import/export trade relationship developed between the two planets or star systems, each would have to make adjustments to their technologies, especially those technologies involved in communication, travel, and trade between the two races. But even other, not directly involved, technologies would be influenced from the likely ripple effect from the contact and trade.
But not only that, another alien race could bring in ideas never yet thought of. They could introduce each other to new concepts, new science discoveries, new data sets, new art forms, and so forth. They would bring in fresh set of eyes (or "eyes"), a fresh way of looking at things - all of which will alter and affect the development of an alien race, from their culture to their technologies.