If your box is that small it is almost certain that it is not a fan rated box. If it is not, then the box MUST be replaced with a fan rated box.
Sorry, but even with approval, someone unlicensed, uninsured, and unqualified SHOULD NOT be doing electrical work in a place that they rent.
This is a perfect example; something as "simple" as changing a ceiling fan turns into a project since the box has to be replaced, and in an older house this can be met with issues.
TL;DR -- find the breaker that shuts that red wire off, then slap the builder with it, so to speak
The builder of your house needs a slap with the NEC. There are two very clear Code violations here, and they're both things that are trivial to avoid.
First off, the lack of panel labeling isn't just a massive inconvenience to you, it's against Code -- the NEC requires it so that the next electrician/homeowner who looks at the panel can actually figure out what breaker to shut off to make a circuit safe or get that sparking oven to quit. In particular, it'd get dinged for busting 110.22(A):
110.22 Identification of Disconnecting Means.
(A) General. Each disconnecting means shall be legibly
marked to indicate its purpose unless located and arranged
so the purpose is evident. The marking shall be of sufficient
durability to withstand the environment involved.
and 408.4(A)
408.4 Field Identification Required.
(A) Circuit Directory or Circuit Identification. Every
circuit and circuit modification shall be legibly identified as
to its clear, evident, and specific purpose or use. The identification shall include an approved degree of detail that
allows each circuit to be distinguished from all others.
Spare positions that contain unused overcurrent devices or
switches shall be described accordingly. The identification
shall be included in a circuit directory that is located on the
face or inside of the panel door in the case of a panelboard
and at each switch or circuit breaker in a switchboard or
switchgear. No circuit shall be described in a manner that
depends on transient conditions of occupancy.
Second, what you have is indeed a Multi-Wire Branch Circuit (MWBC) -- the 52V on the black wire with breaker #23 OFF is stray capacitive coupling from the still-live red wire. The red wire, of course, is still live because of another Code violation, this time of 210.4(B):
(B) Disconnecting Means. Each multiwire branch circuit
shall be provided with a means that will simultaneously
disconnect all ungrounded conductors at the point where
the branch circuit originates.
So, it's time to find the breaker that turns off that darn red wire (leave #23 off while you do this!). Hopefully it will be adjacent to Breaker #23 -- in that case, you can get a handle-tie from the electrical supply house and handle-tie the two dodgy breakers together to fix the issue as per 240.15(B)(1). If the builder was as clueless as I fear though, that breaker is somewhere totally else in the panel -- something you'll want an electrician to fix.
Best Answer
The box you used is slightly smaller (20 cu.in. vs. 21.5 cu.in.), but that won't matter as long as you still comply with the box fill rules. It also doesn't appear to be labeled for fan support, which means it can only support a fan up to 35 lbs. Aside from those issues, I'd say there's no problem with what you did.