Do you have a photo of what is left of your old button?
The handle design "sort of" looks like a 3000 classic typically found in the 1994 and up year range when looking at controllers in which typically uses the long stem button like you found but the modular looks like your year range (91) in which should be using one of the smaller stem button options found in searching the many options of your year range in which some to most of those buttons might not be no longer available while others are once you find your style but if you search the part numbers you still might be able to locate a few places that still have some in stock including Ebay.
This in the following link looks to be your modular design and I believe we ares seeing item number 27 in your photos,
https://www.mercruiserparts.com/bam/subassembly/32078/1989/40
Somewhere in here of the next following link should be your button part number as there are a few to choose from which makes it challenging for me to say for sure this is your part number,
https://www.mercruiserparts.com/remote-controls-and-components-1986-th
Take each button part number and image search it online to see if it looks close to yours / what's ever left of it? You can hover a mouse cursor over each selection to see an image before clicking on each one for a better view, etc..
And here in the next following link are all the ones offered as far as finding parts for most of the Merc OEM shifter components over the years,
https://www.mercruiserparts.com/remote-controls-and-components
Hope this helps narrow it down for you better, good luck.