This partially true. Although there is an incorrect way to blow your nose, there is no agreed upon correct way to do so besides gently. Blowing too hard can result in earaches, nosebleeds, and even the return of some germs into the sinuses. If your nose makes loud noises or your ears pop, you are blowing too hard.
The linked article claims that blowing your nose too hard can result in mucus being propelled back into the sinuses, which can worsen cold symptoms as well as causing nosebleed and ear pain. According to the Allergy Asthma Network, blowing your nose too forcefully can result in germs travelling back into the sinuses. They say if "you sound like a goose", you are blowing too hard. They confirm the possibility of earache and nosebleed and recommend putting a wet washcloth over the nose and forehead to loosen mucus. (https://allergyasthmanetwork.org/news/how-to-blow-nose)
According to an ear, nose, and throat clinic, you can definitely blow your nose too hard. Ear damage is their main concern and they say that if your ears pop, you are blowing too hard. They recommend blowing gently, one nostril at a time, but do not mention mucus thickness. (https://www.harleystreetent.com/blog/our-guide-to-safe-nose-blowing)
Finally, another health page suggests nose blowing frequently as treatment during a cold. However, it again warns about blowing too hard, although it stresses this less than the other sources. (https://www.sutterhealth.org/health/are-you-blowing-your-nose-wrong)
Additionally, it is important to note a lack of information on proper nose blowing from the mayoclinic and the CDC. If improper nose blowing were a large threat, I would expect these websites to mention it. However, there is no reason for the other sources to lie, since they are all health professionals, do not advertise on their pages, and independently presented the same facts.
All sources, including the factchecked article, stress frequent handwashing during the nose-blowing experience. If you are feeling congested, washing your hands thoroughly and blowing your nose gently are the things to do to feel better, although blowing too hard is more likely to result in ear pain than thicker mucus.
https://allergyasthmanetwork.org/news/how-to-blow-nose
https://allergyasthmanetwork.org/news/how-to-blow-nose
https://www.sutterhealth.org/health/are-you-blowing-your-nose-wrong