Is it forbidden to enter the bag claim area through the arrivals gate? Why (not)?

The area where you can pick up your checked luggage is not in any way considered 'secure' and it will not be a security threat if non-passengers would have access. In fact at many (especially smaller) airports, the luggage pickup area is not even separated from the publicly accessible arrival area.

This should be quite obvious, since you have access to your checked luggage, which may contain items which are not allowed in cabin luggage. If you for some reason end up in contact with your checked luggage when e.g. transiting from one flight to another, you will always have to go through a security check and hand luggage screening before you are allowed back into the secure area of the airport.

If I had to guess why the luggage pickup area in most airports is closed to public access, it is most likely a practical issue. The area around the conveyor belts is usually crowded enough if just the passengers are waiting for their luggage and would be even more crowded if all the people waiting for arriving passengers had access as well. Preventing public access is also very likely to reduce luggage theft.

In some airports there is no distinction between arrivals and departures; there is barely a building. This is especially true on small islands where there are no international flights.

(Airport on Taveuni. Foreground: parking lot. Background: a plane. Between: the entire airport, pretty much no walls. Picture by me.)

In larger airports, there is a lot of checking on the "official" way in - security screening, exit immigration processes, and so on - and only people who are carrying a boarding pass for a departing flight and have been screened should be in the inner departure area, while only those who have just arrived should be in the inner arrivals area. The general public is not allowed in either area - sometimes for security reasons, sometimes also for customs and immigration reasons.

Generally this is enforced by signs that say "do not turn back" and even "do not stop" and a person sitting at a desk, apparently doing nothing, who will raise a huge ruckus if anyone "goes backwards" for any reason. Sometimes the entire terminal is evacuated and re-screened. It's a big deal.

In a nutshell, it's often forbidden and there is a sign to that effect. Reasons include maintaining an orderly flow of people and making customs checks easier. At the same time, the luggage claim area is less sensitive than the sterile transit or departure areas so you will find some (typically smaller) airports with doors that don't do much to stop people from entering anyway or even with no separation whatsoever.

In many European airports, where domestic and international (or non-Schengen) passengers use the same luggage claim area, you will find more robust one-way doors before this area and regular sliding doors between the luggage claim area and the public parts of the airport.

Are there signs posted on the doors prohibiting re-entry?
Often, those doors are open so that if someone e.g. needed help carrying luggage, someone could enter to meet the passenger and help pick things up. This assumes it's a domestic arrival area where passengers do not need to then go through customs.

However, most people are going out and you don't typically see people going in, because that is the natural traffic flow for arriving passengers to want to leave the airport on to their next destinations.

Even without customs/immigrations one-way flow, some airports may restrict access to the baggage claim area to reduce baggage theft, illegal taxi drivers pestering arriving passengers about rides, etc. (and some because they are concerned about events like this, but that was committed by someone who just got off the flight).

In general, if there is a restriction against traffic flow, such that you may not enter an area you just exited (e.g. to retrieve a forgotten item or person who seems to have gotten stuck behind), there should be signage indicating the restriction. In the absence of such signage, it is generally allowed to "swim upstream" and go the other way into arrivals; just watch out for colliding with others.

Typically, once you exit the security-controlled zone there is a sign like "once you pass this point you must continue to exit" and then you can't return back without another security screening. However, this is before reaching baggage claim (again, assuming domestic flight without customs).

There are generally three^* sterile areas in an airport. Air Side Departures, Immigration and Customs.

This answer is specifically limited to the difference in Baggage Claim because that is what the OP specifically asked about: "you pick-up your luggage and then you pass through these sliding doors"

For Domestic Arrivals, Baggage Claim is outside all sterile zones so people can freely move in and out of Baggage Claim. Note though, some airports have checkpoints to deter theft where claim checks are verified when entering or exiting.

For International Arrivals, Baggage Claim is within the Customs sterile zone. Once you leave, you cannot re-enter without authorization. While the exit may appear open, it is monitored.

^* Note, I said generally, as in most common. Yes, different airports can have different layouts.

This would vary from airport from airport.

In London airports (for example Stansted) after collection of luggage there is a corridor half way along the corridor there are automatic gates which are always open. Then further at the end there is the arrival exit way no doors.

Between those 2 points there are sensors that detect the movement of passengers, if anyone starts walking backwards those automatic doors get shut. So you are stuck between those 2 points, and will be escorted back by the police to the exit.

I don't know if there are any fines/charges.

It completely depends on the particular airport. I hadn't seen anyone do it in at least 10 years until a few days ago when a women entered with a gift to surprise someone arriving. This was at HUI which is a domestic-only airport in Vietnam.

In large international aiports, the luggage claim area is before customs and it will not be accessible since outside passengers cannot reach customs anyway. In others, it is often blocked by security to avoid theives getting in.

In small airports, including international ones located so small islands such as most Caribbeans and African Islands, the luggage may simply be delivered where the plane lands (so it will be in the secure zone and not accessible) or in a common area which is used by arriving and departing passengers. In one case, this shared area was the parking lot!