A purchase agreement will state whether or not appliances are included in the sale. The bank requires an appraisal on the property. The appraisal will state if built in appliances are included, or not, in the collateral and make adjustments to the value of the house based on that. There are personal items that are often sold with the property that the bank and the appraisal do not consider part of the real estate and typically the bank will not finance those things. A built in stove is considered by both to be part of the real estate and can be included in the financing.
As you said, it “can be.”
Still, unless your mortgage stipulates what you may do (eg, some have a clause about renovations exceeding a certail percentage of the property value), you own the stove and may do with it as you wish.