I have either engineered hardwood or laminated flooring.
This was where my dog's water bowl had been
Is this mold?
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There are three possible ways to fix this.
I don't think you will be able to do option 1 and get a match that will please your landlord.
There are people who specialize in repairing this type of floor damage. It won't hurt to get an estimate. Google "repair wood floor water damage" and you'll get a list of businesses. Someone with experience might be able to match the existing floor.
If the repairing the floor won't please your landlord you might be able to persuade him/her that the floor is old and nearing the end of it's life and you should only have to pay some percentage of the whole cost of a new floor.
I'm sorry I don't have easier options.