Have you tried the cleaner you use for the toilet? My husband does that with my wetroom and it comes up sparkling clean.
If the flooring is linoleum then the cause is most likely not dirt but rather moisture from under the linoleum. When moistures comes up from the subfloor/between the subfloor and linoleum it reacts with the linoleum glue and the linoleum glue seeps up and permanently stains the top surface of the linoleum.
No amount of chemicals are going to remove that kind of stain but, if you don't want to completely replace the flooring, there are linoleum paints and paint sealers available.
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