A raster image inside a vector-based program has a squishy-squashy size. The file will have the same 5300x3500px, but you can shrink it or enlarge it. It will retain the same 5300x3500 pixels regardless of this.
The point that can determine the initial size is the declared resolution. If your image has a declared resolution of 300 PPI ...
Here's a work-around, since PS is a little special in this regard. Right click your photo and figure out its dimensions (on a mac, right click and select 'get-info).
Then create a new file w/ the photos dimensions (e.g., 6000x4000). Drag and drop your imagine into this new file. Now take this layer w/ your photo on it and drag it to whatever other ...
building on @joojaa comment: your artboard appears to be properly set to 1104 px which will, when printed at 300dpi, result in a 3.6 inch printed dimension (1104px divided by 300). I say "properly" because you mention a coffee cup.
When "exporting to 300" your result is 4600px. This means you are upsampling, which usually adds some "...
What should have in a Scalable logo?
I will answer you what it should not have.
It must be 800×800 px
It should not have a fixed pixel dimension.
and 72 dpi.
It should not have a specific PPI resolution.
If you use a vector-based program as Lucian commented, you could start with the default artboard, A4, letter for example.
After finishing the logo, ...
Start building the logo in a vector application, and worry about artboard size and resolution later: with a vector format logo, your artwork will be scalable to ANY artboard size and ANY resolution.
Don't know what a vector application is? Here's a list of the more popular ones:
Alternative to Adobe Illustrator
Is Photoshop a vector application? No.