Sounds like a typical facet joint injury / irritation / sprain whatever you want to call it. The facet joints close when you move into extension and tend to be very sensitive and prone to centralization; this type of injury can linger on for a long time if you allow it to.
Really the only other injury that could cause this type of pain is spondylolisthesis, but that’s rare in your age group (much more common injury in teenagers) and even more rare to only have pain when bench pressing.
Loading the back in extension (i.e. arched) causes the pain and irritation, not necessarily compressive forces in general. This is why squatting / deadlifting don’t really exacerbate the problem, as they don’t really load the spine in large amounts of extension, just in neutral or a little bit of flexion is more typical for those movements.
Here is what you can do:
Sleep on your back; sleeping on your stomach practically guarantees that this will bother you throughout the day. Some people recommend taking this even further and sleeping with a pillow under your knees to take the lumbar spine out of extension completely, but I found this actually tended to be an irritating factor for me.
Only use a tight arch on your top set. You don’t need to get a big arch on warm-ups or even rep work if you are dealing with this type of back pain. For example, if you are following a program like 5/3/1, all your warm-up sets will have no arch (feet set fairly loose) and only on your top set for reps would you move into an arch.
You can still set your scapula to take care of your shoulders, but the arch definitely isn’t necessary for every single work set, especially if you are doing like a 5x10 (or any other set and rep scheme).
I would recommend avoiding doing tons of reps or sets at a high% of your 1RM. If you really want to bring up your bench press #s without torching your back, you can easily do 5/3/1 followed by a 5x8-10 of at 50-60% of your max, and only use an arch for your max set for reps. This lets you get plenty of back training in without irritating the injury site, and you can still make plenty of progress.
Of course, if you are just training for bodybuilding and don’t care about your bench #s, then there’s no need to jam a square peg into a round hole. Seated hammer strength chest machines of various angles work great and won’t irritate this problem. Incline benching also takes a lot of load off the facets and builds the chest just as much (if not better) than a flat bench in my book, since most guys struggle with developing the ‘upper chest’ anyways.
For rep work, you can even bench with your feet on the first rung or even bend your knees and put them on the bench itself (gasp, I know); this will take the load entirely off the facet joints. Same thing when you lay on the ground and do a floor press with your knees bent (though I feel this doesn’t hit the chest very hard or carry over much for a raw lifter as doing full reps would, at least for me).
Strengthen your abs with anti-extension movements; roll-outs, standing abs, etc - anything that strengthens the abs in preventing extension will carry over in a very nice fashion to flat benching.
Do incline bench work; you can get a lot of good bench work in without having to hold nearly as tight of an arch. Mentioned it briefly earlier but it could use a re-mention.
Walk around a bit in the morning before sitting down at your desk / in your car / for the day. Rolling out of bed and plopping down on the computer is not good. Make your breakfast, eat standing up, take a shower - find something to do. Running out of time or I would elaborate on this more, lol.
I had a similar problem once I started working with 300+ pounds on the bench. This now doesn’t bother me and I recently hit a 405 (raw) bench, pain-free.