A symbol is a symbol. The symbol of the Grey Order is a sword, the Grey College offers swordsmanship classes, the EIC symbol is of three swords - Van Hal's, Markus', Mathilde's - Mathilde was taught by her now deceased friend Markus. There's a lot of weight to swords in Mathilde's life, and they've taken on a certain significance.
Other than abandoning the Grey Order, I suppose you could sell not using the sword as a rejection of sentimentality - the axe may be more convenient to get made, so the sword is dropped for practical reasons. Despite everything that the greatsword might mean, Mathilde abandons it due to pragmatism. That's a kind of watershed. Or a repudiation of everything she achieved in Stirland, it could mean that too - she's the last sword of the EIC, so if she drops it...
There's a lot that, symbolically, abandoning the sword could mean, but what it isn't is the equivalent of a generic hair-cut scene - a change for the sake of change.