Everywhere I tried searching 気象庁 on the internet, I found that it is only associated with JMA. But my 先生 firmly argues that 気象庁 does not only mean JMA but general meteorological observatory/agency. I found 気象台 for general observatory but my sensei is adamant so I wanted to hear it from natives. Thanks a lot in advance!
庁 is a kanji for national agencies, so you cannot use 庁 for observatories in general. For example, a weather station run by a university or a private company is definitely not 庁, although it may be called a 気象台 or 気象観測所. World Meteorological Organization run by the UN is not 気象庁, either. Here is the current list of Japanese 庁: https://ja.wikipedia.org/wiki/庁
庁 itself can be used for other countries' national agencies similar to Japanese 庁. For example there is 大韓民国気象庁 in Korea. Still, not all national organizations in charge of weather are run in the form similar to Japanese 庁, so 庁 is not a suitable generic term for them. For example, Italian Meteorological Service is a sub-unit of Italian army, so it's not a 庁.
Japanese Wikipedia has a list of government-run meteorological offices: https://ja.wikipedia.org/wiki/気象機関の一覧 Judging from this and some articles linked from this, there is no simple "generic term" for JMA-like organizations in various countries. You have to say 気象機関, 気象に関する国家機関, etc.