Searching for the name of a rotary switch/push button switch combination
Combined rotary switches with push-activation combinations have never been commodity items that you could buy separately unless you were a major customer. They are virtually always "OEM" components which are custom-made for a specific mass-market product. In previous eras, you could buy "replacement" parts if you found a particular switch you liked in a consumer product, etc.  But in the modern era, we don't believe in component-level repair, so you can't even buy them as "replacement/repair" parts any more.

In virtually ALL modern products the combined rotary/push function is implemented as a ROTARY ENCODER with a push-switch function built-in.  There are many different versions of this available. But they assume that you are using a micro-controller and the switch simply sends "step-clockwise" or "step-anti-clockwise" pulses to the micro-controller.