Some laminate floors are absolutely fine with basic multi purpose, others not (especially if you’re using a bit too much in relation to the water, which is easy to do if you don’t measure it - one I have is just 1 capful/30ml to 5 litres of water)!
White vinegar and warm water can work wonders if a film of residue has built up BUT test somewhere inconspicuous first! You only need to mix a bit up in a jug, say 50ml white vinegar/150ml warm water, then apply with a well-squeezed cloth/sponge. No need to rinse, but repeat until the patch is completely clean. If the shine doesn’t return, it might be that the top layer has worn away?
I use a separate spray bottle and microfibre pad system (as a cleaner, it’s easier this way as I might have 3 different hard floor types in a single house, so changing the bottle would take more time) - as long as the floor isn’t being over-wetted (think grip not slip), and the pads being changed regularly, I don’t find this just moves the dirt around? It’s hard to wring a traditional mop out enough to make it safe for laminate floors, and manufactures recommend pads and spray anyway. On a really dirty floor, I’d spray, mop, allow to dry (which takes no time), change the pad and repeat. Never had a problem this way, and I do a wipe test with piece of kitchen roll to check all is clean.
Stardrops laminate wood floor cleaner is a good budget choice - 99p for 1 litre, also works well on vinyl.