In the DMG, it says that some items are sentient (p. 214).
Some magic items possess sentience and personality. Such an item might be possessed, haunted by the spirit of a previous owner, or self-aware thanks to the magic used to create it. In any case, the item behaves like a character, complete with personality quirks, ideals, bonds, and sometimes flaws. A sentient item might be a cherished ally to its wielder or a continual thorn in the side.
It also says you can make magic items (p. 128).
Magic items are the DM's purview, so you decide how they fall into the party's possession. As an option, you can allow player characters to craft magic items.
So here is my question. Can an adventurer make sentient magic items?
Yes, you can. At the DMs discretion.
The 'Crafting Magic Items' downtime activity (p.129) allow the following (emphasis mine):
Sentient magic items are in the "Treasure" Chapter and in 'Creating Sentient Magic Items' (p. 214) there is an explanation how to make a magic item sentient (not the actual in-game process, which you would have to work out with your DM, but design wise), which implies that any magic item can be sentient, even self-designed ones.
Be aware, that a Sentient Magic item is an NPC/a persona played by the DM.