I always took it to mean something much more than God granting Adam and his progeny the right to keep pets.
Dominion means everything; from pets to food and everything in between.
In the context of BOTH the verses you posted from earlier in Genesis and the ones I gave later in Genesis ... it clearly means to “manage” according to what God told them.
The environment had two MAJOR CHANGES from the time of creation. One major change was to subject the entirety of creation to futility as a judgement at the entrance of sin into man’s world. No longer would it be sustained as it had before, but decay and degradation entered into creation.
The second major change was upon the earth, and it was a drastic degrading of the earth ... at which time the entire environment of the globe was degraded, by God, because of the evil ways of mankind upon the earth.
Thus, to even survive, God gave all animals for mankind to eat. Because of how SEVERELY God judged the earth, because of mankind’s sinful ways - mankind could no longer even survive without becoming meat-eaters.
THAT is how severely degraded our very environment and existence is - as a result of sin.