Is the technique of circular breathing the same across all wind instruments? For example, if I learn to circular breathe on a didgeridoo, will it be easy for me to learn how to do so on a trumpet?
While the mechanics of circular breathing are independent of the instrument, your mentioned instrument examples already have widely varying air pressure requirements.
Without the diaphragm support to generate the necessary pressure (because it is currently busy with inhaling), you have to compensate that with muscles round the mouth. I imagine, that a didgeridoo requires little pressure, while an trumpet requires substantial one (oboe even more). So even if you are perfectly able to seal the trachea against the mouth, the muscles may still be insufficient even for medium passages.
Learning should be easy compared to training the muscles.