What’s your opinion on adults telling children their emotions

That sounds manipulative and would have made child-me very anxious. It’s much better presented as positives, like “I’ve been looking forward to seeing you” or “I’m so happy you’re here”.

Horrendous manipulation to force negative feelings on a child's conscious.

When our GD was about 8 she had a teacher whowould say stuff along the lines of
You make me feel so sad when you've not bothered to learn your spellings, and sometimes you've made me so sad at school I want to go home and cry.
This would be to individuals, groups and while class. When my DiL was telling me I told her to speak with HT immediately and get it shut down. This was emotional blackmail/abuse which turned our GD from confident caring young girl to an anxious mess who was scared incase her behaviour cause her teacher some emotional upset.

You're the child's step mother?

That's not sharing emotions. That's making someone else responsible for your emotions and your reaction to those emotions.

I will sometimes talk to my kids about my age appropriate ways. It's good for them to know that adults feel things too, that it's ok to talk about those emotions and also makes sure they're aware that those emotions might not have anything to do with them. Something like "I'm a bit sad today because I got some bad news about X. I'd really like a hug if that's ok, and maybe we can do quiet things this afternoon." Or "I'm feeling really stressed because of some things that happened at work today. I'm trying to forget about it but I'm finding it a bit difficult to turn my mind to something else."

But the case in the OP is miles from that. I might say "I missed you so much, but I cheered myself up thinking about what a good time you were having" or whatever.

It's fine, and helpful, for adults to use emotional language around kids and to model managing their emotions, but kids should never feel responsible for adult's emotions. It's abusive.

My DF used to do this to me and my brothers as siblings after my DM kicked him out. Every weekend we’d see him, he’d tell us that he missed us so much all he did was cry all week waiting to see us, he couldn’t eat or sleep, and he might as well be dead, etc. I was 5 years old and I still remember the panic every time he dropped us back home that I’d never see him again because he was going to kill himself.

He did it to punish my mother. He wanted us to be angry with her and put pressure on her to take him back. He was an abusive twat.

Yuck. What was the context? Was the child at the other parent becuase then it goes from weird and inappropraite to manipulative as I imaigne it's a way to stop the child wanting to go to the other parent.

It absolutely IS emotional abuse imo. Guilt tripping and no need.

It's usually acceptable to tell your child what's going on emotionally and to name your feelings, and demonstrate working through them.
.

"I'm feeling disappointed because I didn't get the job I wanted. I am going to ask for feedback so I can improve for next time I get an interview, but I still feel a bit down."
"I'm feeling frustrated because my computer isn't working. I am going to go for a walk to calm down and then come back and try again."
"I am feeling very, very sad because a relative died. I might well be sad for a few days and I will need you and Daddy to be patient with me whilst I try to feel better. You might see me crying a lot but it won't be forever and I will be OK."

Talking to your child about your personal relationship issues is very rarely acceptable, and the above examples are just abusive. He/she is just trying to make them feel guilty for visiting their other parent and it's part of parental alienation. In line with the above examples,

"Did you have a wonderful time with Daddy? I missed you a few times and gave your teddy a big hug, but I did lots of fun grown-up things and some boring ones too! I've been thinking of all the fun things we can do together now you're back."

Yeah that’s bad, fast track to screwing up a child.

Yes, exactly - my mum and dad used to say things like 'We missed you! But we had lots to do while you were away so the time went really fast' or 'It's so lovely to have you back home, we missed you. But guess what? While you were away we had time to [do some sort of grown up shopping / socialising thing that I would have found boring]'.

That isn’t sharing emotions, that’s emotional abuse.
I will absolutely share how I’m feeling with my child (e.g I’m a bit grumpy because something happened a work or I’m a bit sad because my friend is poorly) but what is being said in the OP is wholly inappropriate

When I see my kid at pick up from school I say 'hi honey, I'm so glad to see you again!'

I prefer it to 'I missed you so much!' as I don't want them to think it's really normal for someone to intensely miss someone who has done something for a few hours. And I don't want them to feel like going to school is causing me upset because I miss them so much.

Ugh to the examples

Well that's certainly a fast track way to screw up your kid

My friend used to tell her kids she missed them when they were at their dads. Her rational was that she didn't want the kids to think she didn't miss them when they weren't there. When I pointed out to her that it might make them feel bad about going to their dad's and that they would worry about her being alone, she realised it was a mistake. She stopped saying it and if they asked if she missed them, she said 'Of course but I've been very busy getting the house clean and tidy and going to work'! It's not fair to put the emotional burden on the children even when you miss them dreadfully.

That's awful.

I tell DD I missed her - but not in a way that makes her feel responsible for making me sad! I did it yesterday, she came out of school and I said 'hi Baby, did you have a good day? I missed you today' and gave her a cuddle.

This is abusive, it's making the child feel guilt for making the adult sad. It's outside the child's control, and it's not the child's job to manage the emotions of a fully grown adult.

It's counter productive too. I'd want nothing to do with someone who carped on like that every time I saw them!

I'd think the adult was emotionally incontinent and needy and setting that child up to go NC as soon as he or she is old enough to get away.

It's absolutely emotional abuse to make a child feel guilty and sad about spending any time apart from a parent.

It's given the child a clear message that they are somehow responsible for that adult and their emotional state.

That it is their fault that the adult was distressed and crying.

A one off, just bad parenting, constant pattern for their relationship - abusive.

I'd say depends on the context and I'm not sure necessarily abusive. I think the sentiment 'i really missed you whilst you were away' is fine.

'i was crying every day', perhaps makes it the child's responsibility or suggests the adult had lost control of the situation which may cause anxiety.

School trip where the child should be having fun and feeling safe, not fine to make it so drawn out. The adult needs to control themselves and put the child first. Obv they may be sobbing every night but it's not the child's issue.

Maybe if the child was in hospital, more understandable to show more emotion. I'd say if it was something like a non life threatening accident after the fact, fine to show emotion and say they'd had a cry, but maybe a child going through something more worrying might need more reassurance.

This is actual interesting post as I feel it might be cultural. I'm from a very stiff upper lip family. Parents losing control emotionally like that would have caused a lot of anxiety. If it was a culture where crying and emoting were more every day, I'm sure it wouldn't.

Horrible

In the examples you've given, it's not appropriate. It's making the child feel guilty for being apart from the parent, and that's not healthy. Absolutely fine to say 'Yay, you're back! I missed you. Did you have a lovely time?' but 'I cried all day because you weren't here' or 'When you go away and have a lovely time I just sit here crying and being sad' feels very manipulative and suffocating.