Backup all databases in the same place
Consider using Ola Hallengren's maintenance scripts. His scripts are community agreed upon gold standard for this type of work. His scripts will break out each server and instance into separate folders for you automatically. 

Going to one location is fine. Some companies use a Data Domain which a SAN built for backups. You essentially write to a central network share. Writing to a single location for backups is not uncommon. 

Now....some things to consider when doing this


network bandwidth to the device you are writing too
network bandwidth from the device you are backing up
compute power of the device you are writing to


The idea with these items is to think about not overwhelming your backup location. You also do not want to kick off too many backups on one server and use all of the network bandwidth on your server. This would cause timeouts and other client side issues. Maybe break up the workloads by environment. Non-prod starts at 5pm (the end of the developer day), while production starts at 8pm (the end of the business day). This way your backups are not all firing off at all once. You may need to adjust further based on what results you receive. 

Test, test and test more with a small subset of servers. Then you can adjust to scale out for the larger subset of servers and databases.