Full disclosure: am a guy.

Holding your breath allows you to tense up and clench more than you can while breathing. Taking a deep breath eliminates some "tensing" muscles from the equation. Still, with some effort (or anxiety, probably) you can still clench up while breathing.

There are two parts to this. For starters, sex without arousal is also painful. During arousal, you secrete lubricants, the vaginal cavity expands as blood flows to it, and your body dampens pain reception across the entire body. Most people are not aroused by medical examinations and given your examples for experiences I'd say you're definitely not getting hot and bothered by it. So the difference is eating a candy that you're excited for compared to eating plain bread with a dry mouth.

Second is that your doc may be aware of your issue and forewarning you so you're prepared. Given the way you talk about it, I'd imagine you're probably on the younger side. If you're getting your first few exams, or have been direct about your experiences so far then it's likely they're playing it safe for your comfort.

Lastly, talk to your Doctor about this issue, here are some examples so you know the verbage. I'm a guy but I do have second hand experience. I long distanced a gal who had surgery for microperforate hymen before we met and was incredibly scared of penetration in any way. It didn't work out for other reasons, but we had a lot of fun building her confidence and other interests. The mother of my child had a partial transverse vaginal septum, it led to a sharp and buring pain at the entrance during penetration outside of a specific angle. This was "solved" during child birth when the doc had to snip her to allow the baby to come out. After getting stitched back correctly (no husband stitches here) she ended up having quite the sexual awakening. Lastly, my now spouse had a rotated womb which lead to abdominal pain during arousal as it pinched the womb. This was remedied.by a full hysterectomy for unrelated purposes.

Point being, there's a lot of variables in the area and it would do good to read up on some of the issues, do some self exploration to get a good handle on what your symptoms are, and them talk to your Doctor about it.

Because you're not usually sexually aroused at the doctor's office.

Sex is very different from physical examination at a doctor's office. A doctor may need to do things that you wouldn't do for pleasure but are still important for checking on your health. Some of the things they may need to do may be uncomfortable/painful for some people with vaginas. It's also a psychological thing: pleasure during sex/masturbation has a fairly significant psychological component. It's rarely the case that anyone automatically enjoys any interaction with their genitals (male or female genitals, actually). It's very context dependent and dependent on one's frame of mind. Most people are not going into doctor's appointments with any desire or expectation for pleasure, so the experience is just very different from the get go.

Another possible reason for asking you to take a deep breath is that deep breathing can relax muscles and the mind, which can be helpful during an examination.

I would imagine if you visit their office because of pain, a part of their reason and response to you is to brace for the discomfort that you're there for.

If the docor said "this is going to feel good" that would be a lot weirder

I am appalled that no one has brought this up so far- but it hurts because they are touching the cervix, which generally doesn't get touched, and can be very painful to hit. The speculum they are inserting is designed to slightly open the cervix for the examination.

OP, there are many sizes of vagina. You may have one that's on the smaller side, with a low pelvic floor. My partner is the same way, and penetration can be painful for them. We have switched to using very small toys or focusing on clitoral stimulation to get them off instead of relying on penetration.

If your doctor clears you, you may just have a low pelvic floor, and may not like penetrative sex. Which is fine! It's just good to know about yourself.

Because it's not sex, so you're not excited, blood flow hasn't increased, natural lubrication hasn't increased, muscles haven't relaxed naturally.

Several reasons. One, they're sticking a metal thing in there, and that can be painful, especially if they don't use enough lube. I remind them to use a lot. :) Two, you may be having a lot of anxiety and tightening up all your muscles (I know I do!) so that makes it harder for them to insert the thing--hence, they tell you to take breaths and try to relax. Also, I always try to go to a woman gynecologist. I'm more comfortable with someone who knows what it's like and they're usually much nicer, so that helps.

Medicine traditionally does not consider the pain of women or children to be worthy of consideration. They only started using anesthesia on infants in the 1980s, as doctors had convinced themselves that infants couldn't feel pain, for some reason.