Is there anything in 5e similar to the 'catch hold' in 4e when falling? We've been house ruling, but I can't find anything in the core books.
There's no official rule for this, but it's a common enough house rule. In D&D 3e (and AD&D) I allow them to make whatever check is situationally sensible to try to catch them, and then require a Strength check for them to not be pulled over as well. Some DMs seem to feel like it is being "too soft" on the players to give them a second chance, but in my experience, giving them a chance to catch their comrade with a house rule is not unbalancing at all. Because it's not unbalancing to give them a chance, the actual DCs and checks you use don't really matter for balance purposes either.
The implications of this house rule are actually rather interesting. It feels like giving the PCs a break, but it actually just ups the ante on the trap. Instead of having one character fall in on a bad roll, that bad roll triggers a double-or-nothing gamble: neither PC falls in, or both fall in. It's actually rougher on the PCs because the stakes are higher, and it makes it more likely that there won't be anyone at the top who can help them out of the pit.
Hilarity ensues when a third PC tries to stop the second one from being pulled in, and so on. I've had an entire party rope themselves together, forget to tie the rope to anything other than a PC, and all of them fall down a chute to a nastier level of the dungeon due to bad rolls.
There is no beast feature which reduces falling damage
There are no beast or creature features that reduces fall damage. So, you aren't going to get anything there.
If your Wild Shape form has flight there is an optional rule which can help
In Xanathar's Guide to Everything they give an optional rule for flying creatures and falling part of which says:
If you’d like a flying creature to have a better chance of surviving a fall than a non-flying creature does, use this rule: subtract the creature’s current flying speed from the distance it fell before calculating falling damage. This rule is helpful to a flier that is knocked prone but is still conscious and has a current flying speed that is greater than 0 feet. The rule is designed to simulate the creature flapping its wings furiously or taking similar measures to slow the velocity of its fall.
Thus, if your DM goes by such a rule, you will have a way to reduce fall damage if you have a fly speed.
But this only works if you are already flying (or at least have a fly speed) when you start falling.
You can't transform into any creature while falling...
Transforming into a Wild Shape takes an action (or a bonus action if you are a Moon Druid), however in 5e falling happens instantly.
The rule for falling assumes that a creature immediately drops the entire distance when it falls.
Thus, when you start falling there is no time to do anything to reduce your fall damage (unless it is a reaction or readied action which triggers on falling). Once you start falling, you have no chance to use your action or bonus action to transform. So the whole idea of transforming as a way to mitigate fall damage is a nonstarter according to the rules.
...unless you are falling an extreme distance and using an optional rule
The only exception to this would be if you are falling from a height of over 500 feet and your DM uses the optional falling rule from XGE:
When you fall from a great height, you instantly descend up to 500 feet. If you’re still falling on your next turn, you descend up to 500 feet at the end of that turn.