This is an interesting but difficult topic. Are there signs that your cat wants a cat companion? A well-known veterinary website says that there are seven signs your cat might need a feline companion. They confidently list those signs such as clinginess, a change in grooming habits, eating too much or too little, litter box issues, destructive behaviour, unusual sleeping habits and a shift in overall energy. I just don’t think it’s that simple and I don’t agree with this list. There can be many reasons for these behaviour ‘problems’.

Reading your cat
You cannot be clear about your cat wanting another cat! It’ll be guesswork as to whether you believe that your cat would like a cat companion and it is going to be guesswork as to whether a new cat companion will get along with your existing cat. It’s quite a problematic area which a lot of people tend to overlook. This in turn can lead to households where there is a bit of stress between the cats, undermining harmony. I don’t like that thought because one thing you don’t want is for the harmony in the home to be reduced by the presence of a new cat. The whole purpose of having a domestic cat is to improve the harmony in the home.
The big problem
I’ve been struggling with this myself. I’ve been considering adopting a second cat. I look into the eyes of my cat and ask him whether he’d like a cat companion. I believe that many domestic cats would like or benefit from a companion of their own species. They have to adjust to living with a big human. Humans have starkly different characteristics, which is obvious. You’d have thought that a domestic cat would get along with another domestic cat because they have similar personalities.
Most cat owners realise that it isn’t that simple because the inherent trait of the domestic cat is one of a solitary animal. That’s how their wild cat ancestor lives. This solitary nature has been substantially eroded by about 10,000 years of domestication, so we can say today that domestic cats are quite sociable. But, and this is a big but, if you put two or more cats together in a home, and if they are full-time indoor cats, you have a potential conflict over territory. Right away you have compressed the natural size of their home range (several acres at least) dramatically which is a potential issue and you’ve decided for them whether they are going to get along or not.
Chemistry
I believe that domestic cats like and dislike other cats. It’s a question of chemistry just like humans. And we also know that domestic cats have their own personalities. The whole question of whether an incoming cat gets along with the resident cat is problematic and the outcome uncertain. Yes, you can introduce them slowly and they will learn to get along but wouldn’t it be nice if they got along more or less immediately and there was no need for this gradual, mechanical introduction to find some harmony? It seems to be a second class situation if you have to do that.
If you have two fight against the resident cat’s dislike of the incoming cat you’ve failed in my opinion because you’re never going to get true harmony. They will live with each other, work around each other and find their own little perches and hiding places but what you desire is for your two cats to snuggle up together. You want them to enhance each other’s lives. That’s the holy grail of multi-cat homes. How often does it happen?

Signs your cat wants another cat?
I don’t think there are signs that your cat wants another cat. Perhaps the signs are that your cat is suffering from separation anxiety because she is alone all day. So the question is not about signs that your cat wants another cat but that she is lonely because her human guardian is not around at home. The decision to rectify this problem is much easier to make. It does encourage the adoption of another cat or the person can rejig their life to be at home more often. The coronavirus pandemic has certainly aided in that objective because homework is far more commonplace nowadays.
Two better than one?
The basic model is that a couple of cats is better than one cat if they get along well. The difficulty is in achieving that goal with certainty.

Rescue shelters
I don’t know whether it happens but I think that rescue shelters should allow adopters to live with their adopted cat for one week if they have another cat already. They can then find out whether their cats have a really good prospect of living together harmoniously. If it looks bad they can bring the cat back. Perhaps that contractual arrangement already exists in some cat shelters and if so, so much the better.