You are right about the season. Robert Burns's poem, "To a Mouse," takes place late in the fall, after the harvest and just before December. You have probably picked up on clues in the text such as "fields laid bare and wasted," "weary winter coming fast," and "bleak December's winds coming,/Both bitter and keen."
A good way to approach this question is to consider what a traditional view of this season would be. What words come to mind when you think of "fall?" Are they words like "bleak," "bitter," or "weary?" Or do you think of different types of words? Remember to consider the time and place in which Burns wrote his poetry. A Scottish fall may be very different than autumn in Los Angeles, for instance. These considerations will help you answer the question whether he is using the season in a traditional way.
If he's not being traditional, then you must address how he is using the season in an unusual way. How is he inverting or playing with a reader's expectations of "autumn?" For instance, for someone who thinks of autumn as synonymous with Thanksgiving--lots of food and family and warm conversation--this poem could be a very unusual take on the season.
But I think regardless of whether or not the use of the season is traditional or unusual, it is definitely meaningful. Burns evokes a chilly, dark mood to tell the story of the cold, hungry mouse. The speaker, knowing that the mouse's house and food supply have been destroyed by the harvest, allows it into his home and doesn't begrudge it some food, "An odd ear in twenty-four sheaves/Is a small request." As the speaker continues to compare his life with the mouse, however, he finds that there are some instances where the mouse is actually superior. When he says in the final stanza:
Still you are blest, compared with me!
The present only touches you:
But oh! I backward cast my eye,
On prospects dreary!
And forward, though I cannot see,
I guess and fear!
he means that the mouse will soon forget about the dreary fall, while the speaker will always remember hard times in his past and have to prepare for hard times in the future.