Mortgage is a (secured) debt, a combination of a promissory note, and a security interest providing the mortage holder a secured interest in the property. Yes, you are "in debt".
But that depends upon whether you define the term "in debt" as a debt appearing on the balance sheet, or the net of assets - liabilities is less than zero, whether you have a "debt" expense on the income statement (budget), or whether the net of income - expenses is less than zero.
One person might look at their budget, find the (monthly) mortgage payment listed, and judge that they have a debt payment, and thus are "in debt". Or they might look at their expenses, find they exceed their income, and judge that they are "in debt". Another person might look at their balance sheet, compare assets to liabilities, and only say they were "in debt" when their liabilities exceeded their assets.
Some people view mortgage debt as "good debt", as they view certain debts as "good" and others as "bad". Trust me, having a high mortgage payment (higher 30% of your net income) is hard, and over 40% is bad.
Consider you balance sheet and your income statement. On your balance sheet, the house appears on the "asset" side with an (estimated) value, while the "mortgage" (really, the promissory note part of the mortgage) appears on the "liability" side. On your income statement, your house does not appear on the income side, but the mortgage (promissory note) payment appears on the expense side. So, you clearly have both a "liability" with a clearly-defined value and an "expense" with a clearly-defined payment.
But do you have an "asset"?
According to an accountant, you have an "asset" and a "liability". But you do not have a business asset that is producing revenue (income), nor do you have a business asset that can be amortized and expensed to reduce taxable income. When we think about an asset, does the word have the connotation of some thing with value, something that produces income? Well, by that measure, a house only provides income when we rent it out, and only has value when we consider selling it.
As millions of families discovered during the housing (price) collapse, when the market price of your "asset" falls substantially, your personal financial status can fall negative and you can be "broke".