There are, broadly speaking, 4 different possible problems:
The previous oven may have had an electrical problem, but it is very unlikely that a new oven would have a problem. If the breaker tripped immediately when the oven was plugged in or when starting a particular cycle (e.g., bake vs. broil) then that would be a likely oven defect. The breaker tripping after the oven has been on for a while, particularly with a new oven, is likely to be a problem someplace else.
There is a big looming safety issue with the receptacle. It is an older ground-on-neutral NEMA 10. It should be a NEMA 14, with the ground-neutral bond in the oven removed. But that is not by itself going to cause a breaker trip.
Unfortunately, it appears that you have a 3-wire cable without ground. Assuming that's the case, switching to NEMA 14 will require getting a ground from elsewhere, which may be easy or may be hard. You can attach to another ground, but the ground on a standard 15A or 20A circuit is not large enough for a 30A or 50A circuit. That's separate discussion should you be ready to pursue it.
However, there can be problems in a junction box if there are loose or corroded wires. The connections can heat up, arc, start a fire, melt the box, etc. All bad things. So making sure that the box, receptacle and wires are assembled properly is important. But I doubt it is the source of the breaker trip.
The wires need to be able to properly provide the power needed by the oven. The breaker needs to protect the wires. There are some variants depending on wire type and some special cases, but generally speaking the standard US copper wire in cables (as opposed to separate wires in conduit) and breaker sizes are:
- 15A = 14 AWG
- 20A = 12 AWG
- 30A = 10 AWG
- 40A = 8 AWG
- 50A = 6 AWG
In a quick search, it looks like 8 AWG can actually handle 45A. There may be a basis for allowing a 50A breaker on 8 AWG wire, since 45A breakers aren't a standard item, but the actual load of the oven is a key factor. Oven power requirements vary quite a bit depending on size and features.
The issue is that if the breaker is larger than appropriate for the wires, the wires can overheat without the breaker tripping. That is actually the opposite of your problem here, so I doubt it is the wires. However, you should check the size of the wires and the requirements of the oven. If the wires are 10 AWG or smaller then you need to switch to a smaller breaker. Check the oven specifications for the required breaker size. If it doesn't list the breaker size then look at the power requirements, multiply by 1.25 because of continuous usage, divide by 240 (assuming you have 240V power) and that is the minimum number of Amps that your wire needs to provide, and you breakers need to allow, to function properly.
The final piece, which I suspect is the culprit here, is the breaker. You feel heat in the breaker. If everything is running properly, the breakers should not be noticeably warm. That is an indication of either a problem with the breaker itself or, as with junction box problems, the connection to the breaker. The connection to the breaker includes both the wires going into the breaker and also where the breaker connects to the panel.
Turn off the main breaker. Remove the wires from the breaker. Remove the breaker. Look for signs of arcing or other problems. If everything looks OK then more likely the breaker. If there are signs of problems, it may be easily fixable or that breaker space may be unusable. If the breaker is also the wrong size (see above) then replace it with a new breaker. If you're not sure about the problem, it may be best to replace it with a new breaker - typically $10 - $20.