Question edited for clarity.
Question moved to model category.
Yes, you are correct, disconnecting the fuse box has broken the pairing between the key and the immobilizer system. Did you remove the main connection plugs to the fuse box? My thought is that those connectors have never been removed since the car was built and breaking a remaking the connection, you might have some connections that as not as good as they should be. Pull the connectors again and give them a good spray with electrical contact cleaner (that is NOT WD40). Then reconnect.
There is various ideas on resetting the Immobilizer here.
https://www.google.com/search?q=mazda+miata+immobilizer+system+reset
Or a Mobile Auto Locksmith
https://www.google.com/search?q=mobile+auto+locksmith+near+me
Just for fun, here are the Fuse Box locations and diagrams. They have changed over the years, and as you didn't state the year, here is all of them.
Mazda MX-5 Miata (NA; 1989-1997) fuses
Fuse Box Diagrams
https://fuse-box.info > mazda > mazda-mx-5-miata-na...
Here you will find fuse box diagrams of Mazda MX-5 Miata 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, get information about the location of the fuse ...
Mazda MX-5 Miata (ND; 2016-2023) fuses
Fuse Box Diagrams
https://fuse-box.info > mazda > mazda-mx-5-miata-nd...
Fuse box diagram (location and assignment of electrical fuses) for Mazda MX-5 Miata (MX5) (ND; 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023).
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