Hello Cassandra Disiere,
When you check the purchase, was it through the Microsoft Store order history, or through the Minecraft Launcher at minecraft.net directly?
It's important to note that if the original purchase was made directly through Mojang's website before Microsoft's acquisition, it would not appear in the Microsoft Store order history. The Store will always show "Buy" for legacy Mojang licenses regardless of whether the game was already purchased.
The correct way to access a legacy Mojang purchase is through the official Minecraft Launcher — not the Microsoft Store. Sign in with the Microsoft account that was used during the Mojang account migration, and Java Edition should appear there directly.
For installing on Linux Mint, the same Minecraft Launcher from minecraft.net has an official Linux version available for download on the same page.