While the holidays can bring up memories of loss there's also a push for holiday cheer so the pain can wait if only for a little bit. With the holidays over everything that was ignored in favor of being around others can come surging back without a reason. First off I just wanna say that whatever your feeling you are entirely allowed.
Whatever judgment you may receive shoving your feelings aside will only hurt you in the long run. We all sort of know someone who may be dealing with either family turmoil or loss and we want to be there for them in any way we can.
To start with, just checking in with them every few days or so is a baby step in the right direction. If they want to acknowledge their emotional or mental state while talking to you, great but sometimes they just need a distraction.
That doesn't mean you should talk about your own issues. Find common ground or just send them something that'll make them laugh. Make plans keep them busy and engaged if they wanna be. If they don't feel up to doing anything at all but just sitting together that's great too.They might not open up to you but someone else and that's not a reflection on you it's just who they're comfortable with.
If they decide to open up just listen don't ask questions and offer suggestions only if they ask, even then don't be insistent. After don't bring whatever was said up unless it applies to the conversation your having. Emotions are a moment by moment thing and they might not stick around after the person opens up.
If they're saying the same thing as last time it doesn't mean that what they feel isn't real. Be as calm as you can during this as bringing in your own worries will most likely just pile up on them or break where ever the conversation was going. In the end, just do what you can without sacrificing your own health.
To conclude, emotions aren't straightforward and they don't always make sense. They may not even make sense to the person who's having them and you may be able to help them with that. Even if you can't understand their thoughts on what troubles them you can still be there for them or at least not torment them during that time.
Accept that there's only so much you can do for them as well as nothing may be fixed from just talking to you. Neither of you has a psychology degree and even then that can take years of sessions. Just be their friend and hope that that helps in some small way.