Let's start with your last example:
This plant ages quickly in sunlight and that plant ages the same way.
This idiom would seem to be generally acceptable in standard prose. Is it a direct object? I do not think so, because you can use the same way along side a direct object, like her:
He treated her the same way.
He made this bomb the same way he made the other bomb.
This is the most important argument against labelling the same way an object.
Another argument is that it expresses the semantic role of a manner of doing something, as many adverbs do, and it can be replaced with an adverb (e.g. similarly), as you said. That means it is probably best analysed as adverbial.
Thirdly, the verb age normally does not take an object, and you cannot replace the same way with an indisputable object. Only typically adverbial phrases like manner and time work: *she aged an old woman.
I am inclined to agree with Aeismail that it is short for [in] the same way, ellipsis of the preposition. The same applies to that way, this way, some way, the other way, the way [that]... (as in do it the way [that] she does it). The word way lends itself well to adverbial use without a preposition, as do many other nouns expressing a place or time. But somehow in is not normally omitted from in other ways or some other combinations.
Now let's move on to your first example:
I wondered the same thing this morning.
This idiom sounds informal or colloquial to me (though it can also be archaic or poetic), but it is used. How does it work?
What happens when we use the same thing with verbs that typically have a direct object?
I made the same thing last week.
This is possible, and it occupies the semantic role of theme or possibly patient, both of which roles are typically associated with direct objects. With wonder, it occupies the role of theme.
However, like age, the verb wonder normally does not take a direct object. The exception is a direct object in the form of an interrogative subordinate clause, as in I wondered what that mark on the wall was. But you cannot normally use a simple direct object: *I wondered that mark on the wall.
I am, again, inclined to agree with Aeismail that this is ellipsis, of the preposition about. The verb wonder commonly uses about plus a noun to express a theme: I wondered about the mark on the wall, about her, about the cause of the accident.
The question remains why ellipsis commonly occurs with wonder plus the same thing, but not with other noun phrases.
She thought the same thing.
*She thought her mother.
She thought about her mother.
*She thought what this could be.
This seems to indicate that the same thing is indeed special, as you suggested, since it can induce ellipsis (if that's what it is) with other verbs that are normally intransitive too.
Note that think happy thoughts or think clouds, while possible, have a poetic or colloquial ring to them: I wouldn't say it is a normal, standard construction that one would use in, say, an article about Ukrainian demonstrators.
The more philosophical question is whether it is better to call wonder the same thing and think the same thing elliptical prepositional phrases or newly created objects. The fact that the same thing is a bit of an exception, and that it is not possible in formal English, pleads for ellipsis. But both options seem defensible.