How to tell the digital resolution an image?

The replies further down this thread are correct but I added this as it may help your, or your cleint’s, thinking. Apologies for some info which is repeated.
There are three places where where you will see ppi (or often, incorrectly dpi) used:

1. Scanning
In this context it means setting the scanner to record x number of pixels per inch of the document being scanned. In other words convert the physical document inches to a number of pixels. Obviously the higher the ppi setting (on the scanner) then the greater number of pixels are recorded in the document. So at this scanning stage, setting a higher ppi vale in the scanner does affect quality by increasing the number of pixels recorded (there are limits on the capabilities of different scanners before interplolation is used)
The finished document may (or may not) have that ppi setting recorded in the metadata that sits alongside the image data in an image file.

2. Working in an image editor (such as Photoshop)
At this stage ‘resolution’ is simply the number of pixels in the document. In your example an image with more pixels, covering the same subject, will have more information recorded. The ppi value however, just sits alongside the image and is used to set rulers (if they are showing a physical dimension and can be sent to the print driver. It also translates text size to pixels if text is added. For images destined for display screen the ppi  value is not used for anything else and and makes no difference to quality. Hence the comments below that an image of th1 same pixel size with ppi set to 1ppi or 1000ppi does not contain any more, or less pixel information and hence has the same quality of resolution.
In Photoshop, if an image is opened with no ppi value set, then it assigns 72ppi just as an arbitary value to set rulers and text.

3. Printing:
When an image is sent to a print driver, the ppi value is also sent and is used by that print driver to set how many pixels make up one inch on paper. So for the same image, with the same number of pixels, a higher ppi value will make a smaller print but the individual pixels become less visible.
Note : dpi is often used instead of the correct ppi. dpi refers to the dots that make up the individual pixels.

Dave
 

Your example file resolution is 2706x1831 pixels.

Let’s emulate the resolution in a new file to cross-check against.

Create a new image file at 2706x1831 pixels at and assign 360ppi to it. This means that placed objects can be measured against that 360ppi. 

Next I check a Penny Black’s physical dimensions: 

3/4 inch wide by 7/8 inch tall (approx 19 x 22 mm) -->wikipedia

Let's create a new layer at that size with the stamp image from wikipedia: 19x22mm:

If I would print this image at 360ppi on paper this Penny Black’s size would approximate the original size when measured with a ruler.

Notice that the number of stamps in a row equals your example.

This means the example file that you were provided with was indeed scanned at 360ppi. 

When it was scanned, the scan software or software used to prepare the scan assigned a PPI value of 96ppi to the image data.

If you would change that 96ppi value to 360ppi (and be certain to TURN OFF Resample!!!), and print your file the stamps would be the same size as the original scanned stamps.

In short: yes, it is confirmed that your file was scanned at 360ppi and that the pixel dimensions match a 360 pixels per inch resolution which, compared against the physical dimensions of a real Penny Black stamp, is the correct pixel resolution when printed at 360dpi.

This is probably the most misunderstood concept in all of digital imaging.

The fact is, the ppi number isn’t important. All that matters is the pixel dimensions, how many pixels wide by how many pixels high.

A digital image is just an immaterial array of pixels, nothing more. A pixel is a sample point, a data point, it doesn’t have a size.

Now, for many purposes it is necessary to give the image a size. If for instance you want to print it, you need to decide if it’s going to be 15 cm wide or 25 cm wide. That’s where the ppi number comes in, as a way to translate from pixels into physical dimensions. It tells you whether you have enough pixels for the required size.

This is expressed in a simple formula. Ppi, pixels per inch, is the relationship between the pixels in the image and the desired physical size. The ppi number defines the size and vice versa. Pixels per inch. It means exactly what it says.

If an image is 3000 pixels wide, printed at a width of 10 inches, the resulting ppi number is 300. And conversely, a 3000-pixel image with an assigned ppi value of 300, will print at 10 inches.

That same image with an assigned ppi value of 600, will print 5 inches wide. As the ppi value goes up, size goes down and vice versa.

You can assign any ppi number you want, and nothing in the file changes in the slightest. All that changes is how big it will print.

Now, the ppi number can also be used as a secondary parameter for resampling the file. That’s where people tend to get confused, but it’s basically just reverse mathematics, using the formula backwards. It doesn’t change the basic underlying nature of what ppi means.

​@T68203159 DPI is only related to printing, so unless you going to print them you can ignore the DPI, the pixel dimensions are what’s relevant here