The tax issue can be complicated by the fact that in Ohio, prior to 2000, mobile homes were treated as they are still in Indiana. If the home was purchased by whoever the current owner is prior to 2000, they were issued a title (similar to a motor vehicle), and paid a mobile home tax (similar to vehicle excise tax).
Since 2000, mobile homes are treated as real property, and if they're on land owned by the homeowner or attached to a permanent foundation, are charged property tax.
Your issue seems quite muddy as you're presumably the owner of the property, but the actual owner of the trailer isn't entirely clear. Depending on the age of the trailer, you might want to start at either the assessor's office or the BMV to find the actual owner of the trailer. It will be difficult to answer whether this person's property is truly "abandoned" until you know who the rightful owner is and how long ago they last used the trailer.