Do I need a chain breaker tool to install a new chain on my bicycle?
You need a chain tool. All drivetrains are different. The chainring size (or the big ring size if a 2x or 3x drivetrain), largest cog size, and chainstay length all influence the length of chain needed, and all bicycles differ in these parameters. 116 links is a standard length that suffices for the longest required chains on most bicycles. Hence, that length will provide just enough or too much chain for most bikes.
Naturally, you can outsource that task to your local bike store. Each person's inclination for doing their own work and maintaining the necessary tools is different. Changing your chain is easy enough, and many cyclists should expect to do it often enough, that I think a chain tool is a worthy investment. However, your bike store will be more than happy to check for wear and install a new chain, although their workload may affect how quickly they can do it (NB for people happening on this answer in the future: there is a global pandemic now, and bike stores are working overtime since many people want to get their bikes fixed plus they often have to reduce capacity to let their workers stay at a safe distance, so this is not a trivial issue as I type this. We are, we hope, at the beginning of the end of the pandemic.)