I'm interpreting your question that you have this saved image as a picture (raster), with no world file or other georeferencing information. And you say approximate is OK.
In this case, especially given the grid is shown, you could use QGIS'
Raster > Georeferencer feature to force-stretch it to a desired projection rather than to try to guess/determine the projection. (Editing to add: though others' suggestions how to try to sleuth it out are good ones; if you can do so easily that's great. This is if you can't.)
Set your project CRS to something you can work with afterwards, with preference given to your strongest hunch of what the image's projection might be.
Load up from a known data source a simple map that lets you set the canvas extent to slightly exceed the map on your image (e.g. OpenStreetMap, Google, ...).
Vector > Research tools > create grid, create a grid of latitude and longitude to match what you see in your image. Set the Grid CRS to EPSG:4326 to be creating the grid in Lat/Long not in metres, and set the Grid extent to the map canvas you zoomed into in step #2. Turn snapping on for this layer.
Start georeferences, load up your image, and in georeferencer's
Settings > Transformation settings > Transformation type choose
Thin plate spline. Make sure the Target SRS matches your project CRS. (There are improvements in the about-to-be-released version 3.18 of QGIS around georeferencing between different CRSes, but in the current release and LTR versions it is best to have the same CRS.)
Create GCPs along the intersection points of the grid, matching them to the grid intersections on the map canvas. The grid you created there in step #3, and with snapping on, this will be quick.
Hit Georefencer's Play button to create a modified copy of the image, stretched to fit the grid, and loaded up as a layer in your canvas.
A few comments:
A. The more your project CRS is different from the unknown image's actual CRS, the more imprecision there will be in between the grid points. Conversely, if you're pretty darned close (in the same family), you might be able to get away with only 4 points and using
Projective rather than
Thin Plate Spline as the transformation type.
B. This georeferences this one image only. However, if you have other images (e.g. from scraping a web site over time, since this seems to be some radar/weather map) which have the exact same (unknown) projection and extents, you can manually copy over the GCP points and world file you did for one image and use them for the others. Make sure that in your transformation settings, you keep
Save GCP points on.
C. You can of course undisplay or even delete the utility layers of the orientation map and grid in steps 2 and 3 once you are done. You can even reproject your fitted image to a different project CRS on the fly, though image quality may suffer a bit when you do so.