Do you look at the right problem?
You see PCs not set up for your encounters. That might be for different reasons:
- The encounters rely on character classes or feats that are not present? Then the GM messed up not accounting for that. A group of characters with no flying or range capability shouldn't need to fly to get an encounter done. If a lower combat-capability group needs to fight, it might require treating them to be at a lower level than they are or treating the thresholds higher. Sadly the DMG does not address this in encounter design, so you will have to learn what works with your party or make the party composition fudgeable.
- The PCs are very suboptimally built for fluff reasons? Then the GM messed up not accounting for that in encounter design, just as much as the players did by not communicating their dead choices. If the characters are built way below optimal, one can assume them to be of a (slightly to much) lower level than they actually are in planning the encounter.1
- If the characters are just suboptimal due to a bad understanding of the game, sitting together and fixing such problems can bolster a group's capacities a lot by switching a couple of choices The GM's or a senior player's advice can help here a lot.1
- The Players acted badly or had bad luck? ok, NOW the GM is free of fault. If they do stupid things, bad things happen.
Giving them more choices or bolstering the group capabilities can solve the problem, as can fixing problems in encounter design. Just adding a PC might be one of those fixes, but it is far from the only possible solution.
For example, players don't get a full-fledged PC as their aide but a side character. Or the players could get a pool of characters to pick from, but only play one at a time and have to choose before the Adventure starts. That's pretty much a West-Marshes style. And last but not least, in older editions, Gestalt was used to directly bolster the capabilities of characters.
Multiple PCs at the same time is a small tradition
This is a question of style. Nothing actively bans multiple PCs at the same time, but it is the expected norms in many games, that each player only uses one PC at the same time. However, even this is not universal, and not even D&D had that in all editions, especially with side-characters:
- Rangers get an Animal Companion, Druids and some others get access to Summoned Creatures and yet other classes get Familiars or similar feats that represent a creature or being that will join the fray. It's generally considered that as they are part of the character sheet, it's the player that has them on the sheet has to play them. However, some GMs treat that as dictating the intended actions, and at times choose that something different happens than told/expected, or pick up the mantle of such Companion for a little before passing it back.
- Since at least D&D 3, PCs could take a Companion/Follwer trait or similar that would generate a lower-level type of humanoid side character that would follow them, take orders and execute them. These NPCs would often, but not always, be handled by the player they belonged to.
- In D&D 5E, such an additional character is called a Sidekick. Tasha's Cauldron of Everything and has specific rules that explain how such a character is made and suggest some handling.2 While generally, such a character is an NPC, some GMs allow a player to play them in addition to their normal PC, while Tasha even suggests treating that sidekick NPC as a group asset that is ordered around by group consensus.
- The whole idea of "side characters" being nigh mandatory is one of Ars Magica Edition's concepts. It's generally expected each player to bring about 3 or more PC sheets (a Mage, a Companion, 1+ Grog) and switch between which they actively use in the same session as needed or possible. Often the choice would present only a few times an evening, but sometimes the Mage or Campanion brought at least one of their Grogs with them and then would be expected to play both. Grogs are somewhat akin to a sidekick or follower in D&D.
Nothing really prevents it rules-wise, but it is not usually expected to play two fully-fledged Player Characters. Doing so can be quite taxing on both GM and all players, as having multiple present PCs has big implications on spotlight (who a scene is focussed about) and reward distribution debates as well as the logistic nightmare who has which PC where. It just gets complicated super fast. Having a really small group alleviates that, as even the game admits:
Most of the time, each player runs one character. The game plays best that way, without overwhelming anyone. But if your group is small, players can control more than one character.3
Less capable characters, such as Sidekicks or Feat Characters like Animal Companions or Familiars have much less of an impact on the problem of sharing spotlight or new item distribution. They also tend not to be played in a way where they separate from the group on side objectives, which keeps the problem of confusion lower.
Multiple PC at separate times in the same campaign.
There's no rule that forbids a player to have multiple PCs in the same campaign and then only using one each day.
In a West-Marches-style game I ran, with the twist that they didn't even need to leave the town 99% of the time, one player had two PCs I remember best: an elven magic archer focussed on healing & magical DPS and a human officer that was some kind of strong leadership combattant. Which character they used on a particular day was always dependent on the rest of the constellation and the goals of the day. When necessary, they noted what the unused PC was doing those days, and that's it. Simple as that: People showed up, told who they wanted to play with character class, and level, and when they managed to put together a group they thought viable for the task at hand, the adventure started.
If adventures are short and happen from some kind of home base, this needs none to little change in campaign-style but opens the breadth of possible group constellations that can be made, allowing to fill gaps in the setup by swapping PCs as needed to fill the lacking shoes. When not searching to fill a perceived gap in capabilities, striving for specific class synergies can also be a reason for swapping characters. Equally valid for character choice can be to match a given adventure's surroundings, and I quite enjoy it if Players choose their characters in such soft factors.4
Does the adventurer clearly need a person capable of finding traps when its seed is encountered? One of the players has to take out their 3rd level(+) Rogue(Thief), Cleric, Druid, or Ranger with Find Traps for the adventure and have their wizard take a round of research. Is the next adventure starting socially in higher society? Time to break out the Paladin, Bard, Cleric, or Rogue alternate characters instead of the Barbarians that fit such surroundings better.
This puts only almost no strain on each player but for having to maintain one or more alternate character sheets, but allows the group to adapt to various turns and expected encounters.
A tricky question that is opened by choosing from a roster is handling XP. I have seen three styles in use, to various effects.
- XP for only the present PCs. In the best cases I saw, it lead to most players trying to swap their characters more rapidly to keep them equally viable when choosing the characters for the day. This does work in my experience, but if one isn't doing PC swaps, this can open a huge divide between the over-leveled PC and the rest. This can lead to problems if that opens a serious divide between different players' characters: Several PCs can become under-leveled, compared to the rest of the group, and then become a felt "roadblock" to the over-leveled PC's player. This is less an issue with a larger pool of PCs, where a viable party composition can be achieved by figuring out a party that fits level- and capability-wise. This ability to choose becomes a problem with a small pool.5
- XP for the Players to all their PCs. This promotes showing up, as your PCs all come up equally, but it opens a gap to players that have little time and put them at a disadvantage. The only time I saw this, it was abandoned after a few weeks, as log keeping for the GM became a headache. If the group attendance is permanent, this is not separate from the next style.
- XP as a Group statistic. No PC had its separate XP count, every PC had the same XP count, which was on the group log. I have the best experience with this. It resulted in players missing on sessions becoming a non-issue and still offering them (or new players) a way to join in with a character that was on the same level as the others, though they would still miss out on equipment.
Honorable mention: a Gestalt style thing could bolster capabilities... but there's no guidance for 5E
Back in 3.5, there was a playstyle that is known as gestalt. A slightly adapted set of it I encountered in a small Pathfinder group. The gist is, that each PC would get two classes and take all class feats as well as the better statistical modifiers or hit dice. The idea was to bolster a character's capabilities but not increase the number of PC in the party or under one player's control. The D&D 3.5 Rules were never officially adapted to D&D 5E and would need to be adapted by your group.
What is a Character anyway?
Character is sadly a little broader than the querent might have meant it. In this context, I believe it is better to talk about PC, if meaning full-fledged, independent characters, to separate it from NPCs that are not meant to exist independent of a character or place they are made for. While the latter is by default run by the GM, some GMs allow players to roleplay what the assisting NPCs tied to their PC do, because it distributes the load of roleplaying them.
I handle assisting NPCs of PCs in such a way that such an NPC is usually acted out by the player controlling the PC unless I need to convey information through it that the player doesn't have. In combat, such assisting cast characters usually don't have an impact, as most of them retreat. Typical examples of such characters are retainers, servants, handmaidens, acolytes, or clerks that are tied to a PC's estate or business. They don't even manage to qualify as sidekick most of the time and are for flavor and roleplaying a position only.
1 - D&D assumes some kind of mid-to-high character optimization for combat capabilities in the DMG. Achieving this is... tricky. it also is balanced around a mixed party with varying classes and capabilities.
2 - Tahsa's Cauldron of Everything, p.142.
3 - D&D Dungeon Master's Guide, 5th Edition, p.236
4 - A side result is, that the cast of the campaign becomes larger and the whole story of the campaign can tell stories that run concurrent but separate from one another, in case two adventure casts have no PC matching. I encountered this once: one week the attack on the BBEG's dungeon happened to get the McGuffin of the Campaign with cast A and the next week, cast B of the same players ran the assault on the BBEG in his mansion. The conclusion of the two adventures happened with a mix of cast A and B, using the McGuffin on the BBEG.
5 - In games other than D&D, where levels are absent or have little impact, an Experience divide usually is a non-issue, and tracking of XP for alternate PCs separately is what I experienced as the absolute normal thing.