Pretty sure this had an engine swap. The 1KZ-TE didn't come around until 1993ish.
Can you post a picture of the data plate on the firewall?
That said I bet someone swapped the whole wire harness over which included a factory immobilizer ECU:
part 89780-60030:
| 89780 | 89780-60030 | 01 | 04.1996 - 01.1998 | COMPUTER ASSY, TRANSPONDER KEY | KZJ9#,VZJ95..RHD |
Which puts that a 96-98 engine according to the part number. Was probably an alarm/electric door lock setup. The body of yours is an older LJ70 that probably didn't have that stuff so you have an issue trying to get it reset/removed.
I'm going to guess you don't have the Key FOBs that have unlock/lock buttons? If so, it's as simple as replacing the batteries, pressing both lock/unlock keys for 2 seconds
Could be the Toyota System 5000 alarm?
Either way, you are probably going to have to figure out how to remove that immobilizer. Sounds like lots of wire spices so you have some work to do. Get a factory manual for the 1KZ-TE and the 1988 LJ70 and dig in.
I'd start with some serious google:
forum 1kzte immobilizer - Google Search
KZJ78 starting problem
Hope that helps, good luck.