My Vy'keen Weapon Consol operator/controller has offered me a B Class Multi-Tool, with lots of broken slots. Quite early in the game, I found a place where I traded my original Multi-Tool for a Class A Multi-Tool. I don't want to lose my current Multi-Tool, but the Vy'keen Weapon Consol operator is no longer offering me anything else to do. Have I reached the end of that storyline, or do I have to exchange my weapon with it, so the storyline will continue?

Note. My current Multi-Tool already has lots of Upgrade Modules, and I'll lose most of them because they were either "found" or bought with Nanites.

You have reached the "end" of that storyline. From there on, he'll just offer different multi-tools over time. You don't need to take any of them.

Thanks. It's nice to know that I can keep my A Class Multi-Tool, but for ages, it has only offered me the first one it did (mentioned in OP).

Not only Vy'keens but also other races may offer you mutltools, and not only during tutorial. As long as you don't click 'Exchange' you can keep your current one.

I don't recall ever being offered a Multi-Tool other than by the Weapon Consol operator (yet).
I got my A Class Multi-Tool from an outpost on a planet, and "exchanged" the default one for the new one for about 3.4 million units.

Few time ago, a Gek in a planet building during dialogue has offered me one, it was total crap compared to my A-Class 24 slots, I refused of course. I wasn't able to avoid the dialogue.

Sometimes you get offered one from NPCs (although not sure if that changed since Beyond dropped), sometimes from monoliths, sometimes from Ops Centers/Manufacturing Facilities, and sometimes from the distress beacons at crashed ship sites.

NPCs in buildings and space stations will sometimes offer a multi-tool if you give the correct answer to their question/situation.

One that I know that always triggers it is the Vy'keen who demands info on a race. One option requires a certain level of Vy'keen relations. Selecting that option will cause him to offer a multi-tool.

As for the weapons research storyline, if I've already got an A-level multi-tool with 24 slots, I just remove the station after completing the last mission. I could be wrong, but I doubt if he'll ever offer an S-class tool, so there's nothing he could offer that I'd want.