I think if your therapist has a characteristic that you have very negative associations about, such as trauma/gender associations, then that may be something that's too big and it's best simply avoided.
Sometimes, it can work the other way. I never thought I'd be able to see a male therapist because I'm female and have been traumatised by men. I knew intellectually that only a few men are bad, but it was hard to feel that. However, I ended up seeing a male somatic therapist, and that involved touch. Against my expectations, it was actually very healing that he was male. On a conscious and an unconscious level it did a lot to change the way I reacted to men as a result of trauma. .
Apart from that, I think it's always an issue not to let the therapist's identity as a person get in the way of therapy. On both sides - both the therapist and the client need to work this. I say identity meaning to include characteristics like race, their possible biases, judgements and opinions, their personality, their own experiences and their life situation. That's why it's good to know as little about a therapist as possible.
It could be anything. I've struggled the most with thinking about how financially secure my therapists are, compared to me.
I don't think it's necessary for a therapist to have similar experiences in their personal background. I think that can possibly have an unhelpful effect as well as a helpful one. It can be harder to be a "blank book" for your client if you have your own history around something. You might also have sensitivities (triggers, if you want to call them that) that a therapist without that background might not have.
How do you build trust when you are not sure your T can ever understand where you are coming from?
Even if a T has some similarities to you there's no guarantee they'll understand where you're coming from. Just because there's a similar experience or background doesn't mean they have the same reactions or give things the same meaning. Just because someone's from the same cultural group (culture in a broad sense - sexuality, race, gender, class, life choices etc) doesn't mean you'll always think alike or share the same views anyway.
A good T can understand where you're coming from by listening to you and hearing you. It's not a requirement that in some way they "are" you, ie that they have things in common with you personally.