StarringTheComputer.com claims it is a GE 635. I could not find pictures of that particular system, but it is consistent with descriptions in the 635's System Manual and Programming Reference Manual.

Here is a better picture from Serenity:

The GE 600 series were the first multiprocessor systems, and were touted for their multitasking and real-time abilities. It was the system upon which the Multics operating system was developed (the precursor to Unix). It also was used by the USAF to track missiles and by NASA to track the Apollo spacecraft:

GE sold their computer division to Honeywell in 1970, who renamed the architecture the Honeywell 6000 series. Here is a picture of the Processor Control Panel of a Honeywell computer:

Comparing Khan, Serenity, and the Honeywell:

Other than the state of the switches and indicators, the left rack of Khan is identical to Serenity's single rack.

The Honeywell rack has slightly different, more modern styles of switches and indicators. This would be expected from a newer computer. The fonts and colors are different. However, the layout and labeling of components is largely the same.

All three sources use the same indicator colors (off, white, yellow, and orange/red). There is one blue or green indicator in the right Khan rack, but this could be attributed to replacement of a burned-out indicator by the prop/set department.

The Honeywell displays one 36-bit register as a row of 36 indicators. It also displays two 72-bit registers each as two rows of 36 indicators. You can see such a 72-bit display in Serenity, and partially cut off in Khan. (I believe this is the M register, which buffers operands in and out of memory.)

Below the 72-bit display in Serenity and Khan is a table labeled "ILLEGAL ACTION CODES". The table is blurry and seems to have 14 entries. The 635 has 16 different fault modes, 2 which are manual and 14 which are machine-generated. Below this is a set of indicators that includes 5 bits of "COMMAND" and 4 bits of "ILLEGAL ACTION", which are consistent with the 635's fault handling.

Below that is a section of switches labeled "FAULT / STOP ON CONDITION". I can read labels for "ADDRESS" and "COMMAND" which are consistent with the 635's manual traps.

Below this, a hand in Khan turns a rotary switch labeled "ADDRESS STOP". A similar rotary switch appears on the Honeywell (last of the first row of dials; also see enlarged on the right of the photo).

Below that in Khan is a rotary switch labeled "TRACKER DISPLAY SELECT" with five positions. The same switch appears on the Honeywell. I don't find any reference to "tracker" in the 635 documentation, so this may be an add-on for missile tracking systems.

The right rack in Khan shows 3 bits of "TAG" and 9 more bits of "OPERATION CODE". Instructions of the 635 indeed have 3 bits of tag and 9 bits of opcode, which the documentation always spells out as "operation code".

The "FAULTS" next to that match the 635's hardware faults.

The labels within "REGISTER DISPLAY" is not legible, but the number of indicators matches what the 635 calls the "Indicator Register", which we would now call a condition code register.

You can't really tell the difference between the 635 and its sister models by the front panel:

The difference between these models was fewer than 10 wires on the backplane. Field service could convert a 615 to a 635 or 625 or vice versa in a couple of hours if necessary; other than those few wires, the 615, 625 and 635 were identical. https://en.wikipedia.org/wiki/GE-600_series

If this is a custom design, then they spent way too much time making it match the GE 600 / Honeywell 6000 family. My guess is that it was an obsolete but real system that someone was more than happy to sell off. They weren't cheap: