For the Ethiopians and Eritreans, yes. For the Copts, no. For the others, I have no idea.
You will note, I hope, that the one source credited in the Coptic Place article is not a Church source (Butler's Lives of the Saints), and the shorter entry on that page references the Ochtoechos, which is not known in the Coptic Orthodox Church and never has been.
The St. Takla link appears to be some kind of translation from some other source, and some of it makes no sense, like how they give his name in Coptic. John of Damascus never had a Coptic name, because he was never connected to the Egyptian Church to begin with. Surely anyone named John would be Abba Ioannis, and we have several saints by that name who are actually recognized in our Church. St. John Chrysostom is mentioned in the commemoration during the liturgy in Coptic, for instance. John of Damascus is not, and does not have an entry in our synaxarium, and is not included in the litanies of the fathers during the midnight praises, etc.
I don't know how or why the Tewahedo commemorate him, but if I recall correctly he is in their synaxarium. It would not surprise me if they received a sanitized version of his writings and life in their language made by some Ethiopian who had a particular affinity for him and wanted to spread the 'good' parts of his writings among the people, similar to how some Nestorians claim that the writings of St. Isaac the Syrian have been scrubbed of their Nestorianism by the anti-Nestorian churches that venerate him. I don't know.