Shenhe (申鹤) is a Cryo harmony buffer. Her role in buffing cryo dps is similar to Xianyun's role in buffing plunging dps (Xiao).
There were rumors (before 2.4) that Shenhe was originally Cloud Retainer. Interestingly, Xianyun's constellation is called Xianhe.
This is the core of her support abilities. Using her E will buff Cryo attacks by giving bonus damage that scales with her total ATK. This bonus damage is added after multiplying ATK and scaling but is multiplied by DMG Bonus, CRIT, RES, DEF modifiers, etc. $$(ATK*scaling + bonus)*(1+\text{DMG Bonus %})*CRIT$$ The bonus part above scales with Shenhe's ATK (82.18% ATK at Lv10) and applies to Cryo attacks. Yun Jin has a similar buff except the bonus scales with DEF and applies to normal attacks.
The effect of the above on damage is $$1 + \frac{bonus}{ATK*scaling} = 1 + \frac{.8218(\text{Shenhe ATK})}{\text{Character ATK}*scaling} \approx 1 + \frac{1}{scaling}$$ Since Shenhe will usually be built with high ATK, the above simplification assumes that around 80% of Shenhe's ATK will be equal to the ATK of whatever dps carry she is supporting.
This is a very rough estimate and may be more or less in practice. But the general result is that the lower the scaling on the Cryo attack, the bigger the buff, and conversely the higher the scaling (like for one big nuke) the lower the buff. The scaling can be 100-200% for Cryo attacks broken into smaller parts such as normal attacks from Ayaka's infused CA or the ice blade hits from Chongyun's burst. In this case, Shenhe's buff can almost double the damage of the attack (if the scaling is 100% then the buff is 2 times). This kind of buff is more broken than Bennett's buff (which can increase damage around 1.5 times).
To balance this, Shenhe's buff has a quota of 5/7 Cryo hits and only applies to Cryo. So it limits the amount of times the buff applies and for what type of moves, or else Shenhe would just be infinitely broken. This quota of 5/7 hits is usually enough for Ganyu, but dps like Ayaka and Wrio want more than that and thus need C6 Shenhe more.
With one of her passives, her E can also buff the damage of normal/charged/plunging attacks, skills, and bursts by 15% depending on whether it is pressed or held.
If she has 3k atk, and most other dps have 3k atk with Bennett, then her lv10 E gives 82.18% more scaling to cryo damage. If she builds more atk (full atk, 700 atk weapon), she may be able to reach say around 5k atk. Her buff then becomes around (5/3)*82.18=137% more scaling.
This is fine for non-melt dps like Ayaka, but for a dps like Ganyu, having a partner that can enable melt (for 1.5 times more damage) may give a bigger buff than 100 more mv. As a result, Shenhe becomes a better support for dps like Ayaka or Wrio compared to Ganyu. But then why not have both a melt enabler and Shenhe? So a team like Dehya-Ganyu-Shenhe-Bennett or Ganyu-Shenhe-Bennett-kazuha (triple harmony) can work. The real reason we are discussing this at all is because teams usually have a 4 character limit (for abyss, outside some quests that allow 5 characters). Ganyu/Wrio melt usually wants a slot for Bennett (atk buff), a slot for vv shred, leaving only one slot remaining. Either Shenhe or a melt enabler (like Dehya) can fill that last spot. If using Shenhe, then one would need to do something like kazuha Q infuse Bennett pyro for off-field pyro-app to enable melt instead, compared to just simply using Dehya E to do something similar. Meanwhile, for Ayaka (usually played freeze), she usually does have an open spot for Shenhe (as Ayaka teams do not always need Bennett, leaving one spot open). The point is that Shenhe can definitely support Ganyu to a decent extent, but in practice as Ganyu can leverage other harmony characters/melt enablers, Shenhe ends up synergizing a bit more (team-wise) with Ayaka and Wrio (especially at C6).
Unlike Xianyun, Shenhe does not have a cap on how much additive damage her buff can give. But realistically, maxing out artifacts and weapons may give around 5k atk (without Bennett).
Her Q will lower the Cryo/Physical RES of the enemy by 15%, but this is less than Anemo VV, Superconduct, Zhongli shield, and Eula E. With her passive, it will also increase Cryo DMG by 15%. With her C2, it increases the CRIT DMG of Cryo by 15% (a similar effect to C6 Sara, C6 Gorou, or C6 Itto).
C1 gives an extra charge to E. C2 gives CRIT DMG to Cryo. C3 gives +3 E. C4 gives extra damage to Shenhe's E cast, up to 250% which is nuts. C5 gives +3 Q. C6 removes the trigger quota for NA/CA.
Her C6 is pretty broken for Ayaka (or Wrio), but not really needed for Ganyu. The reason is Ganyu's CA is split into a few large chunks, whereas Ayaka's CA is split into several smaller chunks. So Shenhe buffs Ayaka a lot more than Ganyu. Second reason is that Ganyu attacks slow enough that a 5/7 quota is enough to last for the cd of Shenhe's E, especially if Shenhe gets 2 charges at C1. Third, Ayaka freeze usually does not have Bennett, whereas Ganyu (and Wrio) can be played with Bennett melt. In practice, this makes Shenhe give a bigger buff to Ayaka (Shenli) than Ganyu or even Wrio. But since Shenhe has the 5/7 quota, and Ayaka's CA hits multiple times, Ayaka really needs C6 Shenhe to avail herself of the additive buff fully.
Note on c4: each time a character procs her E cryo buff (icy quill), Shenhe gains a skyfrost mantra stack, up to 50 stacks, each stack lasting 60s. When Shenhe uses E, she consumes all stacks, and each stack buffs her E dmg by 5%. This gives her an up to 250% dmg bonus buff on E.
2p Glad/2p Shimenawa/2p VH/2p EO/2p Woods: the chasm domain is really good to farm for Shenhe (as well as XY). Noblesse, Emblem for ER can work. Mixed sets of ATK and ER can work. 4 piece Blizzard Strayer can be done for dps freeze Shenhe.
For support build, full ATK or 2 ATK + ER sands. For dps build: ATK sands-Cryo goblet-CRIT circlet. Substats: ATK, CRIT > ER.
5 stars: Calamity Queller, Skyward Spine, PJWS, Homa. 4 stars: Deathmatch, Blackcliff, Catch, Crescent Pike, Starglitter, Lithic.
She can support Chongyun, Ayaka, Eula, Ganyu, Wrio teams. She is best used as a support for Chongyun and Ayaka specifically. As explained above, she does not buff Ganyu as much as Ayaka or Wrio. Also Eula cannot really use the Cryo damage bonus she provides since most of Eula's damage is physical. Eula can only really benefit from the RES shred on Shenhe's Q and the burst/normal attack bonus from her E passive. Shenhe can also work with Kaeya, Rosaria, Aloy and even other Cryo characters like Qiqi.
She can be used on Physical, Freeze, reverse melt teams. Examples include: