Sort of? The deep breathing part isn't relevant, and you're not asked to do that except maybe if you seem tense and they think it may help you relax. The rest of what you describe would happen as part of a bimanual exam. That consists of:
- Person is laying on their back with feet in stirrups, naked from the waist down, shirt or dress or gown or drape folded up to expose the lower part of the belly if needed.
- Doctor inserts two gloved fingers into the vagina, and uses them to first feel for any abnormalities in the vagina, to note where the cervix is so they have an easier time placing a speculum correctly afterwards
- Then they place their free hand on the belly over the uterus, usually ungloved so they have more sensitivity, and feel the uterus and if possible the ovaries while using the fingers in the vagina to push the cervix up from below, so the uterus is pushed closer to the top hand. They may move the hand inside around to adjust the angle or to try to make the ovaries easier to palpate. (Usually you can only feel them in someone with relatively little body fat, and even then it depends on how their particular internal anatomy is sitting.)
If the person has a retroverted uterus, this might be repeated with a finger in the rectum to push the uterus from a different angle / and to use that finger to feel the underside of the uterus. Or during a rectovaginal exam, again using the internal fingers to push the uterus to a position that can be felt from the belly. That's one time where they may ask you to breathe in and out deeply, as they insert a finger rectally on the breath out - either that or tell you to bear down.
Although all of these exams are becoming less common in favour of ultrasound, either abdominal or transvaginal. Even a highly experienced doctor will miss things using just fingers on the belly, while ultrasound reveals structures and problems that may be impossible to palpate. And more and more the speculum exam / pap smear is replaced with just a quick swab to test for HPV, since that does the same job of screening for cervical cancer while being a lot less invasive. Sad for med fetters, great news for most patients. 😃
=====
As to how it feels... pretty intimate? You are very exposed, the doctor's fingers entering you is a very intense feeling, and as they reach up to palpate the belly they often stand and lean forward to get a good angle, so you are a little caged in by their body and the stirrups. Not in a menacing way, but it puts you in a very submissive position relative to them. I always find my cheeks heating up involuntarily, and I get wet, and then maybe turn a little bit more red as I wonder if they notice that. It can be uncomfortable, too, especially as they try to palpate the ovaries, both because they need to press pretty deep to try to find them and because depending on the time of the month your ovaries may be tender / sensitive to touch. It's usually a pretty brief exam, though, like 30 seconds. The speculum exam takes up most of the time, especially if they need to hunt around for the cervix a bit.