You likely (can't guarantee it) have a series of mistakes that happened to work out OK in this particular instance.
The typical setup of wires from two different breakers coming to a duplex receptacle is to have one breaker for the top receptacle and one for the bottom receptacle. Technically this can be for a bunch of reasons, but the usual reason is a Multi-Wire Branch Circuit or MWBC. An MWBC lets you use three wires (normally black, red, white) in one cable to power a pair of 120V circuits instead of using two separate black/white cables. The advantage is that you can power twice as much stuff while saving on wire compared to two totally separate circuits. However, an MWBC only works correctly if the two breakers are on different phases/legs. That can be done with a typical double breaker or with two side-by-side single breakers with a handle tie (because common shutoff is required to avoid exactly the problem you found of a single breaker being turned off not cutting all power to the yoke) or with the proper set of tandem breakers (which gets a little tricky and varies by panel - basically the middle pair of a quad set).
However, in ye olden times an MWBC did not have to be a pair of breakers or fuses next to each other and so mistakes were made. I actually found one when (with my electrician) replacing old fuse panels with a new breaker panel. A common situation is that an MWBC will get rearranged in such a way that the two breakers end up on the same phase/leg, which I believe is what happened here.
But there is a second issue: separating the receptacles. The tabs on the sides are supposed to be removed when two receptacles in a duplex are powered (MWBC) or operated (half-switched) separately. But again, mistakes are common and that's how you end up with situations such as this one, and also one of the ways you get a switch that doesn't seem to do anything because it is switching one receptacle but power never goes away because the tab is there and the unswitched hot powers both receptacles.
And then the fun part:
If you rewire an MWBC to be one one phase/leg and then replace a receptacle that had the tabs removed without removing the tabs on the new receptacle then you get exactly what you found here.
On the other hand, if you replace a proper MWBC duplex receptacle without removing the tabs then sparks will fly until the breaker (normally very quickly) trips.
There is a remaining mystery though: Normally an MWBC is about saving wire, so you typically have a /3 (black, red, white) going to at least the first receptacle and then it might continue that way (with tabs removed on hot side but not on neutral side), or it splits at some point (negating some of the savings) into two /2 cables, each with hot from one breaker and sharing (at the point of the split) neutral. But the reasons to do that is if, for example, you run an MWBC to the first location in a kitchen and then split to other receptacles - but if you do that then you wouldn't have those two /2 cables in one location except at the initial split.
So something else may be going on. The next step is to carefully remove the front cover of the breaker panel so you can see whether this is a /3 from the pair of breakers (in which case it is a very serious issue because that neutral can get overloaded) or two separate /2 cables (in which case the mystery still exists but things area little bit safer).
Based on the picture and breaker identification, these are both black wires. So presumably two separate circuits, not an MWBC. Why? I have no idea. But be careful about disconnecting either one, as it is possible that one could be paired with a red wire on one of the breakers a few spaces down. Need to trace out the wires to the top of the panel box where the cables come in to see what goes with the reds.
As for the existing receptacle problem, capping one set of wires and using an ordinary (tabs not removed) receptacle would be perfectly safe. But figuring out what is going one is still a good idea. Step 2 (step 1 is to check the cables at the breakers) is to see what other receptacles, lights, etc. are powered by each breaker.
