Is in Visual Studio an equivalent to the Eclipse "Run Configurations"?

You can create multiple configurations just like in eclipse, screens are different but you can do pretty much everything. You can click on Menu -> Build \ Configuration Manager and in every combo box, you will see last two options as New and Edit, which will let you copy existing configuration and create new ones. Just next to Run button, you can also change active configuration and once you have changed your active configuration you can also change individual run and project settings with different command lines and different directives for compilation.

Resharper does support run configurations. See https://www.jetbrains.com/help/resharper/Run_Configurations.html

Sadly, you cannot build a run config that starts multiple projects -- something you can do with the (single) run config built in to VS. That would be a useful feature since I'm a constantly switching between a single project run and a multi-project run setup. And although not a huge pain, it is a small PITA.

Since you can easily pick a project to start via the built-in VS run config, I don't see much value in the resharper feature.

Well, you can do some things with a resharper run config that VS run config does not support. You can run an arbitrary exe -- something you can setup via a VS external tool. And, you can run a static method. Sounds cool, but I've never had need to do that.

So, there is a tool that provides the feature you asked for. But as often is the case, the feature may not be all that useful :(