If you go to the right page of the Pokemon's summary, you can check its ribbons, and in Sw/Sh as well as S/V, you can select one of the ribbons (or marks) it has to be pinned. Then any time you send it out, except at the start of a multi battle or S/V double battle, it will have a title added onto its name, which depends on the particular ribbon you chose to pin.
"the Reliable Partner" is the title associated with the Partner Mark, which you randomly get over time by building up friendship and having the Pokemon walk a lot outside of its ball.
Other titles you can get from pinning the various ribbons or marks that are easily obtainable in Paldea:
"the Once Well-Trained" for the Effort Ribbon, obtained by filling up the EV banks to the global budget limit, then talking to an NPC in Levincia;
"the Great Friend" for the Best Friends Ribbon, obtained by grinding up the friendship meter similar to the Partner Mark, but then instead of being random with every step, for this one an NPC in Cascarrafa gives it out when you seek them out specifically;
"the Paldea Champion" for the (gasp) Paldea Champion Ribbon, obtained by beating the league if you haven't already done that, or the Academy Ace Tournament if you have;
"the Gourmet" for the Gourmand Mark, which is sometimes given out after you make a sandwich.
