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Can I install a back bike rack without attachment to the rear part of the frame?

Also, if someone can clarify, whenever I google

bike rack

I get hits on racks designed to carry bikes on a car and such.

I am referring to the racks on the bikes themselves which will allow me to haul things on my bike.

I bought a bike recently assuming it would have attachments near the rear part of the frame but they are not present.

I have seen some bike racks which secure only to the rear wheel axle, but that does not seem too stable.

As you can see in the pic there are no holes for screwing into like I see in many other bikes and nearly all of the racks I looked at.

Several manufacturers make replacement seatpost collars with integrated rack mount holes. Just make sure you get the right diameter for your seat tube. Here is an example:

A search of "bicycle cargo racks" led me to what you are looking for. You can modify your search to specify "seat post mounted racks". There are also models that clamp to the seatstays ( the frame member that goes from the axle area to the seatpost). Just check to make sure the clamps are compatible with your frame type. Clamp on racks are generally a no-go for carbon frames.

Is the seatpost carbon? If it is replace it with an aluminium one because you'll get a number of racks that clamp to the seatpost. A rack that clamps to the seatpost would normally take up to 10kg. You may find one that also has stays that attach to the axle. Some are attached with bands to the seat-stays. This should only be done with a metal frame not a carbon one.

There are various models that clamp onto the seat post (assuming it's round metal). Topeak calls theirs "beam rack" for example.

These tend to have a lower weight limit than conventional designs, but are still more than adequate for many uses.

There are bicycle racks which mount directly on the rear hub which ar completely stable and effective.