Can a home owner kick out a renting tenant for no reason and give them only 2 months to move?

If your family has a month to month tenancy, not a lease for a specified term, then yes, either you or the landlord may generally terminate the tenancy for any or no reason with proper notice, which sounds like this is what occurred in your situation. This is the big difference between a lease for a specified term and a month to month tenancy. In fact, the law allows for as short a notice as one month, rather than two, if the tenant has been living in the residence for less than one year. There are some exceptions, of course. For example, some jurisdictions are subject to local rent control laws which may provide for a different notice period. Also, the landlord may not utilize an eviction as a retaliatory act and discriminatory evictions are unlawful also. If you believe that you have facts that show that your landlord is either retaliating against you or discriminating against you, then you should contact an attorney immediately.

Renting and leasing are the same thing. I presume you are on a month to month tenancy. That means you are renting for a single month but at the end of the month the original rental agreement is extended. If you live in California, at any time you can tell the landlord you are going to move out after a 30 day time expires. And he can tell you the same thing [60 for him if you have lived there for a year or more]. No reason has to be stated by either side. If your city has rent control, there may be different requirements. He owns the house and does not have to rent to you; that is what being a tenant is.

If you have a periodic tenancy (month to month) of over a year then you get 60 days notice, yes.

Unless the property is subject to Rent Control, a 30-day notice is usually all that is needed.