Is it okay that my Sony TV power cord replacement has a slot for a third prong that the original doesn't?

Is this safe to do?

Yes.

Does the third prong of the new cable interact in any way with the other two?

The third pin is a protective earth connection, some appliances "class 1" require a protective earth connection to maintain single-fault safety. Others "class 2" do not.

The connectors are sensibly designed so you can use a cord with an earth for an appliance that does not need one, but not vice-versa.

What are these cables called?

The connectors are from the IEC 60320 series. The connector with an earth is a C13 while the one without it is C17. The corresponding inlets are C14 and C18 (normally the inlet has a number one higher than the corresponding cord connector).

Some of the more common connectors in the series have colloquial names, for example the C5/C6 is commonly referred to as "cloverleaf" or "mickey mouse", while the C7/C8 is commonly referred to as "figure 8". The C13 in my experience usually just gets referred to as an "IEC connector" or sometimes as a "kettle plug" (though this is ambiguous because actual kettles use the "hot condition" C15/C16). The 16A variants are usually referred to by their current rating "16A IEC connector".

I'm not aware of a colloquial name for the C17 connector though. In my experience it's a pretty rare connector. Most class 2 electronics are low enough power to use the "figure 8" connector and stuff like heaters tend to have fixed power cords.

I also suspect that the compatibility mentioned above means there is virtually no end-user market for C17 cords. Regular C13 cords are available everywhere and the economies of scale mean they are cheaper than C17 cords despite having higher material costs.

To answer the question of what these cables are called, they are IEC 60320 (commonly called kettle plugs) connectors. If you take a look at the "Appliance_couplers" section of the linked page you will see the left is c17/c18 (the male and female versions have different numbers) while the right is c13/14.

It looks like from the table the plug on the right is for class I appliances while the cable on the left is rated for class II appliances. In general electrical connectors are built with physical safeties such that they cannot be used in situations where their use would cause damage (this of course doesn't apply to hack-job scenarios like if the end-user were to grind out the safety tab to use a c13/14 connector when a c15/16 is required)

Selection of a coupler depends in part on the IEC appliance classes. The shape and dimensions of appliance inlets and connectors are coordinated so that a connector with lower current rating, temperature rating, or polarization cannot be inserted into an appliance inlet that requires higher ratings. (i.e. a Protection Class II connector cannot mate with a Class I inlet which requires an earth); whereas connecting a Class I to a Class II appliance inlet is possible because it creates no safety hazard.

CLASS I CONNECTORS

These appliances [Class I] must have their chassis connected to electrical earth (US: ground) by a separate earth conductor (coloured green/yellow in most countries, green in India, USA, Canada and Japan). The earth connection is achieved with a 3-conductor mains cable, typically ending with 3-prong AC connector which plugs into a corresponding AC outlet. The basic requirement is that no single failure can result in dangerous voltage becoming exposed so that it might cause an electric shock and that if a fault occurs the supply will be removed automatically (this is sometimes referred to as ADS = Automatic Disconnection of Supply).

A fault in the appliance which causes a live conductor to contact the casing will cause a current to flow in the earth conductor. If large enough, this current will trip an over-current device (fuse or circuit breaker (CB)) and disconnect the supply. The disconnection time has to be fast enough not to allow fibrillation to start if a person is in contact with the casing at the time. This time and the current rating in turn sets a maximum earth resistance permissible. To provide supplementary protection against high-impedance faults it is common to recommend a residual-current device (RCD) also known as a residual current circuit breaker (RCCB), ground fault circuit interrupter (GFCI), or residual current operated circuit-breaker with integral over-current protection (RCBO), which will cut off the supply of electricity to the appliance if the currents in the two poles of the supply are not equal and opposite.

CLASS II CONNECTORS

A Class II or double insulated electrical appliance is one which has been designed in such a way that it does not require a safety connection to electrical earth (ground).

The basic requirement is that no single failure can result in dangerous voltage becoming exposed so that it might cause an electric shock and that this is achieved without relying on an earthed metal casing. This is usually achieved at least in part by having at least two layers of insulating material between live parts and the user, or by using reinforced insulation.

In Europe, a double insulated appliance must be labelled Class II or double insulated or bear the double insulation symbol: ⧈ (a square inside another square).

Insulated AC/DC power supplies (such as cell-phone chargers) are typically designated as Class II, meaning that the DC output wires are isolated from the AC input. The designation "Class II" should not be confused with the designation "Class 2", as the latter is unrelated to insulation (it originates from standard UL 1310, setting limits on maximum output voltage/current/power).

The OEM cable does not have a ground or earth connection.

The cable on the right does.

So this will be a safe replacement for the original cable.

The ground connection will not be used and that wire in the cable is not connected to either of the other wires in that cable.