I am adding a new answer instead of editing as my original answer still is my recommendation since I don’t know your local AHJ. The 2020 code has updates on MC/AC bundling, with focus in 1 area of code usually other methods are also more closely examined at the professional level and DIY more so.
Code allows cables to be run together but because of inductive heating there are penalties that reduce the ampacity of the wires (in cables or not)
We use table 310.15.B.16 for most residential wiring the use of this table is limited to not more than 3 current carrying conductors.
Code requires small wires current to be limited per 240.4.D
However these above requirements have minor exceptions, with exceptions come exclusions and additional requirements.
Table 310.15.B.3.A adjustment factors (commonly called derate) the code covering these covers many pages. And several code articles.
Derating is allowed in residential but the wiring and components must be listed for use with the higher temp tables 75 or 90. And you can not exceed the maximum current rating for small wires 15,20 & 30 for 14,12 &10 awg wire.
Now most equipment is rated for 75 deg so a quick calculation of 15 amp for a 14 awg branch circuit on 75deg or 20a x .8 = 16a so that will work so 4-6 conductors is fine.
Going to 90 degree table the switches, receptacles and load center (breaker panel have to be 90 degree rated). unless commercial equipment is used a residential 15 amp circuit is limited to 3 cables or 6 current carrying conductors. Why not use the 7-9 conductors with the 70% derate .7 x 20= 14 amps so the OCPD (breaker) would have to be smaller than 14 amps.
Ok now for 12awg wire or cables the same rules apply 4-6current. carrying conductors using the 75 deg table for 12awg is 25a x .8 = 20 amps
In both the 12 & 14 awg methods you can see if we could use the 90 degree table that would allow another 5 amps on each wire size and the insulation can handle this load, yes there are load centers out there 90 degree rated and most motor loads (like attic fans and circulation fans are 90deg rated and could use this method) but most residential components are only rated to 75 deg and the inspector will flag more than 6 CCC through a bored hole.
If you really want to go more than 3 use a short conduit nipple less than 24” with bushings on each end then you can have a max fill rate of 60% and No derating is required. This is how we make it down the wall with a 2” or 2-1/2” pipe into the load center with many cables. Remember to calculate the area of a flat cable the width is used.
There is an additional method to do use the 90 degree table but unless a specialty home with many pumps and fans it is really outside DIY as I said I have battled this one with inspectors, and for a few extra holes it’s just not worth it so a short nipple with bushings although expensive may be the best method with limited space. But in this case I see no reason to have more than 2 cables if wanted go to 3 but unless all components are 90 degree rated the inspector can flag the job requiring an extra inspection.
Once you upset an inspector be prepared for them to take out a “magnifying glass” and try to find something else. Note some of my inspectors are also structural inspectors if they feel you are trying to cheat the code they will find things.