Player characters may hold titles in factions. Honorific titles from a faction are simply reflections of how much trust the organization has for someone - they confer no direct power (e.g. to command NPC soldiers or spend faction coin) in and of themselves. With that said, the titles can be assumed to have a certain amount of influence. An officer's word goes a lot farther than a new arrival's, with both faction NPCs and the typical informed Skulker.
Rising within a faction means accomplishing their goals. You can do this either by your own initiative, or by going after specific goals provided to you by NPC leaders. A faction superior's recommendation goes a long way towards membership. If your character has done something for a faction, make sure to submit a Plot Action ticket so DMs know what you've accomplished.
It is possible to lose standing in a faction, either because your PC is perceived to have made a mistake, or even due to intrigues outside of your control and comprehension. A change in the NPCs that lead a group may result in a shakeup of which PCs are favored and trusted. In a chaotic web like that of Skullport, few things last forever, and nothing slips away quite so quickly as trust...
The power brokers of Skullport do not look kindly upon the practice of holding titles in multiple factions, though it might be possible for a spy to conceal their true employer if they are clever. Likewise, the two major drow houses operating here don't recognize claims of Lolthite rank bestowed outside of their House shrines.
The factions bestow on their most trusted members a special level of prestige and influence. Faction superiors are told more about the faction's goals and aims. A faction will almost always come to an officer's aid, but expects unquestionable loyalty and competence. A few titles considered to be superiors within their factions:
House Lysaen: Yathrin, Weaponmaster, First Wizard, First Assassin
Dark Dagger: Priest of Vhaeraun, Envenomed Edge
The Xanathar: Lieutenant of the Hands, the Mouths, or the Runners
Promenade: Dark Lady, Hand of the Protectors, Darksong Knight, Master Harper
A faction will formally recognize and accept those who have served them well. Faction members are expected to show respect to their superiors, even if they are not bound by duty to them. The factions tend to come to their members' aid, though they will expect their members to do the same when called upon. A few titles considered to be full members of a faction:
House Lysaen: Yath'abban, Sargtlin, Faern, Velg'larn
Dark Dagger: Acolyte of Vhaeraun, Point of the Dagger
The Xanathar: Agent of the Hands, the Mouths, or the Runners
Promenade: Maid of the Promenade, Protector of the Song, Harper Agent
Allies and friends of the factions won't typically hold titles, but might still be well-known and relied upon by the full members. Factions will not typically rush to the aid of their associates without expecting something in return. It will not necessarily be obvious who is and isn't faction-aligned.