If you are in India - then no, there is no legal requirement for you to register yourself as a freelancer. However, if you are directly involved with Govt. contracting then you must hold a valid Govt. issued license. As of your income, you must pay your income taxes regardless. If you are getting paid directly to your bank account then it would be good to find out if it is TDS i.e. Tax Deducted at Source. This means that appropriate taxes were deducted (and paid on your behalf to the Govt.) before the monies were paid to you. If it is not, then it is your responsibility to pay your taxes at the end of the financial year (i.e. March for India).
In Other Countries:
For freelancing in other countries, you will face numerous restrictions (almost 95% countries in the world) due to immigration reasons. The reason is that freelancing is considered to be self employment and those countries allow self employment for citizens or naturalized citizens only.
For example, if you move to America in a work/non-immigrant visa (H1/L1 etc.) you will and must be employed by an employer at all times and get monthly paychecks. Also, based on the kind of non-immigrant visa you hold, you may or may not be legally allowed to change employers (example L1 visa holders cannot change their L1 visa holding employer whereas an H1 visa holder can). You will not be allowed to do any kind of work that can be considered as self employment until you obtain naturalization through an immigrant visa application which could take 5 to 10 years (for Indians and Chinese citizens specifically, since they are exempt from visa lottery) depending upon various situations. There are various ways an immigrant visa can be filed and in your case the most probable method would be employer based, where your employer would need to file your non immigrant work permit or naturalization.
European countries also impose several such limitations to non-immigrant applicants. That is, until you obtain a legal immigrant status you would be limited in terms of self employment. European visas are points based and there are numerous criteria to fulfill to gather appropriate points for filing a visa (just an FYI)
As of your question, once you immigrate to a different country (i.e. become a naturalized citizen) your records will be tracked by their Labor Departments automatically since your application for immigration will most likely need labor department's approval (better known as "DOL" or "Department or Labor/Labour"), so there may not be a legal need for you to register yourself as a freelancer/self employed thereafter. However, some countries may indeed require you to register yourself with an additional public or private licensing body for you to be able to legally perform freelancing or start a business.
Although, a small subset of countries such as Bhutan, Nepal etc. may allow you to work there as freelancer without any necessary documentation but for a limited period of time (This applies to Indian Citizens only).
While it is not legally required for you to be registered as a Freelancer in India, it may not be a bad idea for you to be associated with a more larger professional group of freelancers. If you intend to move to another country and continue freelancing, and since freelancing is treated as self employment, you might need to acquire appropriate work authorization or even naturalization or citizenship of that country through an immigrant visa process or other legal procedures authorized by local laws.
NOTE: I am not an attorney or their representative. These are legal topics and you should consult an immigration attorney or paralegal for all your immigration and non-immigration questions and concerns. The content of this answer is just for your information purposes.
Immihelp Comparison of H1B to L1 US Visas- (URL)
Immihelp article on immigrant visa - (URL)
UK's points based immigration system- (URL)
An Economic Times article on Freelancing in India - (URL)