This is a great question and I can see where the confusion might be. I will take you through several steps to solve this logic problem.
First, you need to establish the relationship of Earth and Mars. This is the most important part. Earth is a planet and Mars is a planet as well. So, from a logical and relational point of view, both are planets. We are comparing to like things.
Second, we need to look for an analogous relationship with the word, "Australia." In light of this, we need to determine what Australia is. It is a country and a continent.
Third, we need to see if there is another country or continent among the answer choices. What make this tricky is that the answers (a) and (b) are traps. The correct answer is (c). Asia is a continent like Australia.
In short, the key relationship is that two like things are being compared. Earth and Mars are like things; Australia and Asia are like things. For the former, we have planets. For the later, we have continents.
The answer to this problem is Asia.
For this answer to be correct, however, we have to define Australia as a continent, which may be problematic depending on which geography book you read. To fully and clearly discern the correct answer then we have to take one step past the logic of the analogy and discuss some geography.
Australia is often not described or defined as a continent in and of itself. Rather it is part of the continent known as Oceania. This title describes a unique continent in that it is not exactly "continental" in the traditional sense.
Most continents are constituted by a single land mass, which may also possess some associated islands. For instance, North America is a continental land mass that encompasses Puerto Rico, Manhattan and the Florida Keys.
Oceania is a conventionally described as a continent made up entirely of islands (if we refer to Australia as an island, which is debatable) and includes New Zealand and Micronesia. But we see a terminology issue when we turn to the dictionary for a definition of Oceania.
"...the lands of the cen & S Pacific including Micronesia, Melanesia, Polynesia (including New Zealand), often Australia, & sometimes the Malay Archipelago"
The phrases "often" and "sometimes" appearing in this definition of Oceania should make clear the idea that this continent is special in that it is especially loose in its definition when compared to many other continents.
Turning to the analogy at hand, we should note that Earth and Mars both belong to the same category (planets) and each are specifically named (and not referred to by category). Thus, in the options provided in the answers, we should be looking for a specifically named item. As Asia is the only specifically named item it is the only possibly correct answer.
Australia is, certainly, a country and is possibly and island and sometimes referred to as a continent. Each of these descriptions are possibly accurate, yet in the logic of the analogy we are looking for a one-to-one, direct correlation between one specifically named geographical entity and another, making Asia the only possibly correct answer.