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You are basically considered a tenant and the mortgage payments would be regarded as the equivalent of rent since you live in homes. Absent some written greement identifying an ownership interest there is nothing legally that will create an ownershop interest in the homes for you.
Please be aware that my answer is not legal advice, it is merely information and no attorney client relationship has been formed. You should always contact a local attorney for legal advice.
Yes, you are basically a tenant in the house. It may be possible for you to take some of the improvements with you if they are not legally "fixtures." You are entitled to take what furniture you have paid for. Decorations, art, that are not fixed to the structure would also be items that you could take if jointly paid for. Since these items are not fixtures, purchased jointly you have as much rught to them as your partner.