Is this silent treatment from my 13 year old abusive?

Your young child, who is upset the family pet has been rehomed for whatever reason, is not abusing you, no.

She’s not abusing you. She’s being a stroppy teenager. Parent her, make it clear that whilst you understand her upset, her behaviour is not acceptable. Do it without using histrionic language that awards her more power than she in reality has.

Edit
I cross posted with your update.

Yes that does make a big difference. The cat has more intensive needs that you couldn't meet.

Give her some time and don't plead with her. Let her have her feelings.

She's 13, upset and sulking. She's got absolutely no power or control and had to watch helplessly as her mum rehomed a cat, saying it's a time issue when she probably knows full well that cats are not like dogs and don't actually require walking and are happy to be left at home all day. So she's probably angry and confused about why you lied about the reason for rehoming the cat.

I wish my cat just needed food and water plonking down in front of him

Yes I can leave him all day, I've even left him over night but he does need attention, worming, flea treatment, boosters. He's a bit past hunting now but there is dealing with the 'presents' cat sick and fur balls

Sulking for 24 hours I ignore; beyond that is a consequence but to be honest normally the ignoring it and talking to them as if they’re talking to you works.
I don’t know why you gave your cat away so likely your child is struggling too

I think the real issue here is that you are projecting your feelings about your ex onto your child. You need to process those feelings first -your child is not your ex.

You sound like a victim OP.

You made a decision, now act like an adult and own it. Your dd will come round when she feels like it. It's OK for her to be upset.

How much trouble is a cat?
You are the one who has behaved badly,
Poor kid.

From a 13 year old I wouldn't class as it abuse. She is still learning how to effectively deal with social conflicts. I do think however that at 13 she definitely does need to learn that giving someone the silent treatment is not an acceptable way to deal with a conflict. And you can tell her that if an adult did this it would be classed as abusive.

Don't beg but do explain to her why it's not acceptable. Maybe do the old acknowledging her feelings (missing cat, etc) and brain storming with her how to find a solution.

Well I have to say I wouldn't think very highly of someone who gives up a pet

Cats are hardly time consuming

If I gave our cat away because I couldn't be bothered to look after it, my dd would probably hold it against me for life.

As she should.

Three issues here that I see:

1. You are projecting your stuff onto your 13 year old child. Get yourself into therapy, she is not your ex!
2. You are begging your child to be okay with the situation!!! Be a parent and tell her this behaviour is unacceptable, and had to stop. She can be upset but not talking to you is not the way to deal with it.
3. Cats are not a lot of effort, I'm at a loss to understand why you couldn't cope with it! But in glad he got a good home.

She’s just being a normal difficult teenager who’s upset about a decision made by their parent. Nothing unusual and nothing abusive.

Don’t beg your child to be ok with you because it puts responsibility on her for your feelings, and as a child, she is not responsible for your feelings, you are.

She’s sulking, not abusive. That said, she needs to behave herself and resume speaking with the rest of the family.

I would try to understand her upset though, a cat is not particularly hard to look after and she probably can’t understand how it got to the point where it was rehomed.

Surely this is a joke.

Your 13 yr old is pissed off because you gave away a family pet, not being abusive. I think you need to chat this through with her when she’s ready not accuse her of being abusive. It’s not her fault your ex was dreadful.

Incidentally it’s pretty poor to give pets away especially when you have kids- it really affects them emotionally (even if the cat has gone to a good home). And unless there is a massive drop feed I really don’t see how a cat can have been that much work that you weren’t expecting- surely all that’s involved is putting some food down once a day and maybe a litter tray if the cat isn’t allowed outside?

It doesn't matter your reasons, you've made the decision that you think is best for you and your family. Now stick with it and don't cave. She will get over it if you don't pander to her/fan the flames of her silence. Sulking is no fun on your own when everyone pretends you aren't and just gets on with life. Don't make this into a bigger deal than it is.

A cat doesn’t need someone with it all day.

I think YABU in getting rid of your child’s pet, and to be honest overreacting and overthinking her behaviour.

You have upset her and she is reacting to that.

It doesn’t sound ‘abusive’. Reads like something from one of the narc parents threads.

She's 13. This is a teenager exercising what power they have - it's completely and utterly normal! Words like abuse and triggering is just ridiculous in this context! I'm not entirely understanding quite how time intensive a cat can be that you need to rehome it...?! If you don't want it, you don't want it. Just be honest about the reason.

Abusive? You're overreacting.

Just sounds like very, very normal typical young kid who seems quite sad about the situation (understandably I'd say as in my experience cats are no effort at all but just my opinion).

How much work do you need to do to take care of a cat? YABU for giving it away as cats don't need care!