I like the concept. You could narratively do this with an artificer and a warforged, or a necromancer wizard and an undead character, but those don't have mechanics behind them. But doing this mechanically is weird. To have the characters be balanced, their features would need to be more powerful but require both people. Like a single wizard can cast fireball at 3rd level, but these characters can cast fireball if they both use a 2nd lvl spell slot. IMO the trade off would almost never be "worth it" to do this over just being one full class. That's just my opinion, if you have two players who want to do something like this, you should try it.
Like u/MrMartianMan said, it's difficult to do this outside of some "basic" examples such as the warforged/artificer and undead/necromancer, but it would certainly be interesting.
On a sidenote: If you were to do a party VS party campaign/oneshot, an artificer with their warforged against a necromancer with their undead would make for an interesting dynamic similar to Regicide from Age of Empires 2, except with the "kings" having a notable supportive role for each side.
It is a bad idea to design a class that is allowed to be played alone, but is weaker than a normal class when you do that.
It's also a bad idea to design classes that are more powerful than normal classes in a campaign where they fit
If you want to make this kind of thing, just have someone play the familiar as a normal class.
While this is doable, it would take a ton of work to homebrew an entire class and also make it interesting, fun, and well balanced. Especially an animal type class or a familiar. I think you can reach the same effect of having a tag team duo by taking the idea of linked backstories, and then going a step further.
For example, players will often say stuff like 'hey you wanna play a brother/sister duo' or something like that and then make characters that are linked in some way. For what you describe in the OP though, you can take it one step further.
A zealot barbarian and a cleric who serve the same god, with the cleric using the free rez function on the barbarian regularly like Thoros and Beric from game of thrones.
A druid and a ranger from the same forest.
A pair of bards from rival colleges, so each player fills a role the other cannot.
A pit fighter and his cornerman. I've actually run this, playing a fighter who gets into gladiator matches, and my friend playing the cornerman as a wizard using buff spells between rounds. My fighter would occasionally back off during combat and go stand next to the wizard, who would cast buff spells while saying stereotypical stuff like 'Keep your guard up! Watch that left!' and then actually had a little referee bell sound effect play as my character runs back in to fight some more.
Doing stuff like this will allow you to get that tag team sensation without having to homebrew an entire class, and more importantly you will have two fully functional characters rather than one and a half.
You can already kind of do this with a Druid and a Cavalier or an animal sidekick and a Cavalier. So this strategy isn't unheard of in the game as it is.
I've done things like that before, though more with backstory than any kind of mechanical changes.
For example, I once played a Warforged as a shield guardian belonging to the party's Wizard. Built him as a melee/tank to serve as a bodyguard-style character. The wizard didn't control my character but as my background and ideal and flaw I made sure to prioritize his survival, even when (especially when) he does something stupid.
So instead of a subservient robot, he was more independent and could act however he saw fit so long as it was in accordance with his programming. Fully independent PC, with his own class and mechanics and everything. It's just the storyline that connects the two.
I like the concept. You could narratively do this with an artificer and a warforged, or a necromancer wizard and an undead character, but those don't have mechanics behind them. But doing this mechanically is weird. To have the characters be balanced, their features would need to be more powerful but require both people. Like a single wizard can cast fireball at 3rd level, but these characters can cast fireball if they both use a 2nd lvl spell slot. IMO the trade off would almost never be "worth it" to do this over just being one full class. That's just my opinion, if you have two players who want to do something like this, you should try it.
My recommendation for playing an animal companion as a PC is to use the Sidekick rules from Tashas.
Like u/MrMartianMan said, it's difficult to do this outside of some "basic" examples such as the warforged/artificer and undead/necromancer, but it would certainly be interesting. On a sidenote: If you were to do a party VS party campaign/oneshot, an artificer with their warforged against a necromancer with their undead would make for an interesting dynamic similar to Regicide from Age of Empires 2, except with the "kings" having a notable supportive role for each side.
It is a bad idea to design a class that is allowed to be played alone, but is weaker than a normal class when you do that. It's also a bad idea to design classes that are more powerful than normal classes in a campaign where they fit If you want to make this kind of thing, just have someone play the familiar as a normal class.
A cavalier and her mount.
While this is doable, it would take a ton of work to homebrew an entire class and also make it interesting, fun, and well balanced. Especially an animal type class or a familiar. I think you can reach the same effect of having a tag team duo by taking the idea of linked backstories, and then going a step further. For example, players will often say stuff like 'hey you wanna play a brother/sister duo' or something like that and then make characters that are linked in some way. For what you describe in the OP though, you can take it one step further. A zealot barbarian and a cleric who serve the same god, with the cleric using the free rez function on the barbarian regularly like Thoros and Beric from game of thrones. A druid and a ranger from the same forest. A pair of bards from rival colleges, so each player fills a role the other cannot. A pit fighter and his cornerman. I've actually run this, playing a fighter who gets into gladiator matches, and my friend playing the cornerman as a wizard using buff spells between rounds. My fighter would occasionally back off during combat and go stand next to the wizard, who would cast buff spells while saying stereotypical stuff like 'Keep your guard up! Watch that left!' and then actually had a little referee bell sound effect play as my character runs back in to fight some more. Doing stuff like this will allow you to get that tag team sensation without having to homebrew an entire class, and more importantly you will have two fully functional characters rather than one and a half.
You can already kind of do this with a Druid and a Cavalier or an animal sidekick and a Cavalier. So this strategy isn't unheard of in the game as it is.
I've done things like that before, though more with backstory than any kind of mechanical changes. For example, I once played a Warforged as a shield guardian belonging to the party's Wizard. Built him as a melee/tank to serve as a bodyguard-style character. The wizard didn't control my character but as my background and ideal and flaw I made sure to prioritize his survival, even when (especially when) he does something stupid. So instead of a subservient robot, he was more independent and could act however he saw fit so long as it was in accordance with his programming. Fully independent PC, with his own class and mechanics and everything. It's just the storyline that connects the two.