How to format a partition inside of an img file?

You can access the disk image and its individual partitions via the loopback feature. You have already discovered that some disk utilities will operate (reasonably) happily on disk images. However, mkfs is not one of them (but strangely mount is).

Here is output from fdisk -lu binary.img:

Disk binary.img: 400 MiB, 419430400 bytes, 819200 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
...

Device           Boot Start    End Sectors  Size Id Type
binary.img1            2048 819199  817152  399M 83 Linux

To access the partition you've created you have a couple of choices

1. The explicit route

   losetup --offset $((512*2048)) --sizelimit $((512*817152)) --show --find binary.img
   /dev/loop0
   

   The output /dev/loop0 is the name of the loop device that has been allocated. The --offset parameter is just the partition's offset (Start) multiplied by the sector size (512). Whereas --sizelimit is the size of the partition, and you can calculate it in the following way: End-Start+1, which is 819199-2048+1=817152 , and that number also has to be multiplied by the sector size.

   You can then use /dev/loop0 as your reference to the partition:

   mkfs -t ext4 -L img1 /dev/loop0
   mkdir -p /mnt/img1
   mount /dev/loop0 /mnt/img1
   ...
   umount /mnt/img1
   losetup -d /dev/loop0
   

2. The implicit route

   losetup --partscan --show --find binary.img
   /dev/loop0
   

   The output /dev/loop0 is the name of the primary loop device that has been allocated. In addition, the --partscan option tells the kernel to scan the device for a partition table and assign subsidiary loop devices automatically. In your case with the one partition you also get /dev/loop0p1, which you can then use as your reference to the partition:

   mkfs -t ext4 -L img1 /dev/loop0p1
   mkdir -p /mnt/img1
   mount /dev/loop0p1 /mnt/img1
   ...
   umount /mnt/img1
   losetup -d /dev/loop0
   

There is another way to do this in general, use kpartx (not kde related)

sudo kpartx -a binary.img

and now you should have all partition devices defined under /dev/mapper as loop0p1, loop0p2, ...

and then

sudo mkfs.ext4 /dev/mapper/loop0p1

Optionnaly, when you are done, you can run also

sudo kpartx -d binary.img

to get rid of the loop0p? deivce

I don't know why it looks for binary.img1

(… and later for binary.img2 buried in the commentary.)

That is because the tools are expecting the filenames to follow a specific pattern. That pattern is the one used by device files for actual discs and disc volumes on your system, namely:

• A device file encompassing the whole disc is named sda (or something else). This is what fdisk expects to make use of.
• Device files for individual slices of the disc, described by its partitioning, are named sda1, sda2, sda3, and so forth. This is what tools such as gparted expect to make use of when they tell mkfs to do things on individual disc volumes.

Of course, ordinary files don't overlap in the manner that disc devices files do. The discussions involving the loopback filesystem that you have seen are all about taking a single whole-disc image file and using loopback to create the 1, 2, 3, and so forth files that reflect the individual slices within it, once the desired partition layout has been written to the partition table.