What database and setup can handle several million inserts per minute?
Ok, let me start simple: My old trusted 5 year old main db server (2x quad core AMD Opteron 2378) handles around 80,000 inserts per second, though small ones. That is without SSD for log and data on a SQL Server.

There are plenty of databases that can meet your requirement - it is MOSTLY a question of appropriate hardware. Heck, if you would go Oracle, then the smallest Exadata in a performance configuration (pure SSD) likely can do 10 times as much - we got in a project 5 gigabyte/second stable IO out of it, with a paltry 3 storage nodes.

That said, you WILL have to test it yourself. A lot depends on your programming or lack thereof. If you hit 1 insert per transaction, that is VERY different - and a lot harder to do (not at all because your log partitions need to handle 5 million IOPS then or fake the flush).