DOTA2 does indeed support both, didn't remember it supporting D3D11 - there doesn't seem to be many benches, but the one YouTube video comparing the renderers I did find showed similar performance between D3D11 and Vulkan, not better on Vulkan
I've found DOTA 2 to run a lot more smoothly in Linux with Vulkan than OpenGL. It seems to me that a lot of people don't understand exactly why Vulkan and DX12
can be faster.
First of all, there is very little about Vulkan and DX12 that will just make an 3D application faster, it provides you tools to help accelerate performance. Back in the days of DX11 and OpenGL, most of what the API was doing was reliant on global state and individual draw calls. This means that everything is intended to work together and be indivisible which is great from a stability and predictability standpoint but, is terrible from a throughput perspective. There are some thing in Vulkan that make it a little different such as removing a lot of error checking and reducing overhead to interact with the API but, there is one fundimental contstruct that they added that can yield the best performance:
Command buffers. The nice thing about command buffers is that they're like a queue for things like drawing. Now, there are still some limitations to how many commands can be issued at any given time but, a big benefit is that multiple threads could be preparing command buffers to be executed. This helps decouple the preparation of APIs for things like drawing, texturing, and lighting from the actual act of making them happen.
If there are Vulkan or DX12 ports that merely wrap DX11/OpenGL and literally execute a command buffer for every draw call, then yeah, performance is going to suck because the developers didn't know what the hell they were doing because using Vulkan the same was as OpenGL/DX11 defeats the purpose of using Vulkan/DX12 in the first place. If you're going to be so lazy just to wrap the new API, then you should just keep using DX11 or OpenGL.
Khronos was very clear that Vulkan was not intended to replace OpenGL or OpenGL ES. Vulkan is a harder to use alternative that provides lower level access and greater ability to improve performance. It is not a drop-in replacement for existing 3D APIs.
tl;dr: Vulkan and DX12 are not intended to be drop-in replacements for DX11 and OpenGL and people (as well as devs,) need to stop treating them as such.