I have a server binary (gitea) which I am trying to deploy on google cloud App Engine. I can run it successfully on the google cloud console, checked it using web viewer provided in the console. I have few doubts regarding the deployment on google cloud, which I am listing down here.
Is there any difference between running a server in console versus deploying it? If there is a difference how can I deploy the binaries on google cloud App Engine?
It's generally not as simple as plonking a binary on a VM. Not because this approach doesn't work but because you want to consider security (access|control) and maintainability too.
I recommend you spend some time reading Google's documentation and continue to learn-by-doing to familiar yourself with Cloud Platform.
Some pointers:
There are multiple compute services for you to choose between on Google Cloud Platform (Cloud Shell, App Engine standard or flexible, Compute Engine, Kubernetes Engine, Cloud Run, Cloud Functions etc.). Each, mostly (!) addresses a well-defined use-case.
App Engine has 2 flavors: standard and flexible. Using Flexible, you may deploy a container containing your gitea binary. But, you will need to use a container (and I believe gitea provides one) that includes the binary. See custom runtimes.
Cloud Shell is a Compute Engine VM but it's intended as more of a development tool. Yes, you can run binaries on it and yes you can access these remotely but this is likely not the best solution for you in this case.
Alternatively, (but still containerized) you may also be interested to review Cloud Run, Kubernetes Engine (probably overkill in this case).
Lastly, you could create a Compute Engine instance (VM) and run the binary on it. This is a valid solution too.