Do bigger animals have more testosterone?
It is not that easy. Off the top of my hat I can try to pinpoint a few fallacies in your theory:


Scaling of the body happens in three dimensions and thus to a power law. As such allometric equations have to be applied when dealing with the total amount of substance in the body. Different species have different correlation factors of body mass to total substance amounts.
Angiogenesis (the formation of blood vessels) is affected by Allometry (the relationship of body size to the 'biology')
Diffusion factors are similar in closely related species. This means in order for messenger molecules with a short acting range/half-life (e.g. nitric oxide), the total amount in the animal may indeed be likely higher and the messenger in question may be produced on-site (e.g. endothelium). Testosterone as the name indicates, is however produced in the testes and released in the blood stream.
The sensitivity and regulation of the receptors themselves (which react to a substance) can be different in different species. It may also be weakly responsive to begin with at the genetic and ultimately the folded protein level. A good example of that is the melanocortin receptor MC1R in a gross section of the white human population. Even if you were to produce the same level of melanotonin, the melanocytes wouldn't respond to the same degree as in Humans with darker skin.
Messenger molecules can be re-purposed and take over or be given additional roles and functions over evolutionary-time. Again, Melatonin may serve here as an example.
After all a given messenger is already evolutionary tested, especially in regard to cross-talk, Haptens, and other adverse molecular interactions in the animal species and comes with the full set of molecular machinery adapted to it, - a "luxury" a new messenger doesn't have. As such pressure usually exists to re-use given messengers.



  Like, in animals, does testosterone correlate with size, possibly
  explaining why a lion is heaps more slender, muscular, fit, agile,
  aggressive, and vicious than a typical human?


I doubt you will find typical humans or typical human populations from a biological perspective. Your view may be skewed by economic factors.


  Basically, does the bigger size and aggression in wild animals
  correlate to their higher testosterone, such as larger size equal to
  larger testosterone in comparison to other smalelr or less aggressive
  animals, or is it different?


To a large degree the behavior is determined by the niche the animal is in. As to which concerted molecular messengers make the "evolutionary cut", by best influencing an animals behavior in its given "time-domain" is another question. As the question is complex much is still researched.

Take this counter-example: A mouse and an elephant have very different masses resulting in different inertial forces. Fleeing from an enemy will take different strategies in regard to the central nervous system as well as the body, and demand different levels of messenger molecules at different places, at different times for both animals. Yet the substances in question will likely be similar.