What recourse do I have if travel agent (Gotogate) claims airline can cancel my ticket, but airline asks me to call my agent, which charge a fee?

Most likely, none. Your contract is with the travel agent, and it's standard practice in the industry that they are your point of contact for any changes you wish to make. When the agency says that you can contact the airline, it doesn't really mean much other than that the agency won't stop you from contacting the airline. It doesn't guarantee that the airline is willing to handle your request directly.

From an airline's point of view, reducing staffing requirements by shifting these requests elsewhere is one of the attractions of dealing with travel agencies. Some airlines may still allow you to contact them directly for cancellations or changes, but many do not or will have limits on the kind of changes possible.

If you do want to try again with the airline, you may have more luck contacting them by phone rather than using the website. The staff may have a little more leeway than public-facing websites. But even then your chances will vary with airline policy, and perhaps with any frequent flier status you have.

Ultimately, accepting the possibility of cancellation/change fees imposed by the travel agent is a trade-off you make when booking via a travel agent.

I'll share the outcome of my discussions with airline and the travel agent. During my third call to the travel agent, the employee I had on the line told me that she "made the request to allow changes for my flight ticket via the airline". This seems to have done something because previously on the airline's website, I could see that I couldn't change or cancel my booking. After she made that request, I couldn't access anymore my booking on the airline's website: the website was stuck at the stage trying to find the information for my booking, after entering my info on https://www.singaporeair.com/en_UK/sg/plan-travel/your-booking/managebooking/:

I tried to call again the airline but after waiting for over 3 hours for someone to pick up the phone, I opted to only wait one hour to reach the traveling agent and pay their fees.

Tldr: The travel agent might have the option to allow the airline to make changes to the booking, which the airline website might however not support.

I'll also add that the claim by the travel agent "You can also contact the airline directly to cancel your tickets" whereas that they know very well it is not possible (to the point that they have the option to request that cancellation and changes can be made directly with the airline) is likely illegal in some countries, e.g countries with decent consumer protection laws.