So if a person wants to so make just a fake account on social media. But it is not a real account. Is this considered haraam? Is doing things illegal haraam?
Is doing illegal stuff haram?
In general, yes... unless the "legality" of a particular issue violates Islam. e.g., Niqab (France) and Ramadan fasting (China) are illegal in some countries but does that mean they are haram? No! Plus, alcohol and drugs are legal in some countries so does that mean it is Halal too? No! So it depends on how aligned with Islam something is.
make just a fake account on social media. But it is not a real account. Is this considered haraam?
Making an account that is anonymous is okay but using a "fake identity" (e.g. claiming to be someone else or pretending to be the opposite gender, etc) is haram because it's a lie/deception. Furthermore, if you use the anonymous account just for browsing and other general things it is okay. If you're using it to cause fitnah (trouble) then it's haram. Both the outer demeanor and the intention/actions matter.
Hope this helps. This is a good question. Keep seeking knowledge for Allah's (ﷻ) sake, may He reward you. Neither asking nor answering is ever a waste of time when done sincerely. This is per Islam, not the Dunya.
O you who have believed, obey Allah and obey the Messenger and those in authority among you. And if you disagree over anything, refer it to Allah and the Messenger, if you should believe in Allah and the Last Day. That is the best [way] and best in result.
Quran Ch.4, Surah tul Annisai V.59
In this verse, Allah gives us an order that we must respect Him and His Messenger(PBUH) and in addition to that, those in authority.
Being in countries where the laws are not made by Muslims can be a tricky situation and in some cases it insights some sense of rebelliousness. Islam orders us to respect and follow orders, but in universally agreed we can defy if the laws are against the laws of Islam, you know the general.
It is illegal to drive through a red light but it is not haram, at least not stated in any of the Islamic laws. In accordance with the verse, we are supposed to obey the lights as are stated by the laws of wherever we live.
Back to your question, fake social media accounts, if it is just a different name from what it states in your official documents I see no harm in that. The harm comes in when you use the fake account to do illegal things and or things that hurt other people. IMHO.