I would say that Go has had an enormous and profound impact on artificial intelligence, both narrowly in the application to other games, and much more broadly in how it has affected research and development in many other fields.
Historically (before 2016, in the pre-AlphaGo era), Go had remained unconquered by computers and was widely seen as the “most difficult” of the classical board games. This has served to inspire artificial intelligence researchers for decades. The application of Monte Carlo Tree Search (MCTS) towards Go, beginning in the 90s, eventually led to bots achieving strong amateur dan status, which helped to elevate the status and motivate further development of MCTS and other reinforcement learning topics.
AlphaGo further built upon MCTS by also incorporating neural networks, and has widely inspired a lot more work in AI and reinforcement learning. This success was probably pivotal in granting the DeepMind team the robust support to continue thoroughly extending their line of work into other games, such as with AlphaZero - Wikipedia (Chess, Shogi, Go), MuZero - Wikipedia (adding Atari games), and AlphaStar (software) - Wikipedia (StarCraft). However, more important (to society) are the extensions and applications of such AI methods beyond just games. For example,
Besides what DeepMind/Google has continued to do, and besides other developments on game AI (such as OpenAI Five - Wikipedia, Pluribus (poker bot) - Wikipedia, among others), which undoubtedly gained inspiration and support upon the accomplishments of AlphaGo, are the broader impacts that the AlphaGo milestone had upon the wider AI research community, especially in the area of Reinforcement Learning (RL).
After 2016, so much RL work cites AlphaGo as an inspiration and motivating influence, and the applications span many diverse fields like engineering design, healthcare and diagnostics, self-driving cars, drug discovery, finance, robotics, and industrial automation. To add a personal anecdote, I would estimate that attending a recent research talk on work involving RL has yielded at least an 80% chance of seeing AlphaGo mentioned in the motivation and background section, to the point that it even seems cliche.