No, if they are booked separately, it is your sole responsibility to be there in time for the check-in and boarding deadlines of the second flight. If you miss it, you'll either have to change the booking (before the check-in deadline, usually, though it may depend on the fare, and possibly with penalties/fees and a fare difference), or book and pay for a new ticket. Remember that in some cases, missing the first flight on a return ticket may cancel the whole return trip. Also remember that if the next available flight is the following day, you'll have to pay for your hotel, meals, transportation, etc.
Whether you get to the airport on foot, by public transport, in your car or via another flight does not matter, and the fact that it's the same airline does not matter either.
The airline may take the situation into account, but that's purely goodwill on their part, they have absolutely no legal or contractual obligation to do so.
You may however be covered by your travel insurance (including those sold by airlines and booking sites, or those bundled with some credit cards). This will depend on the exact terms of your travel insurance (and if you have one and it covers this flight, of course).
Options for you include:
- Booking the whole trip as one ticket (multi-city or with a stop-over, in your case)
- Booking the second flight with a flexible fare that allows free or cheap changes (but you're still on the hook for hotels etc. if you have to stay overnight)
- Adding more buffer (overnight is usually more than enough for short-haul flights or flights on busy routes, but may be an issue with infrequent long-haul flights, which are probably a lot more frequent these days).
- Having good travel insurance (double-check the terms and exclusions).
Good travel insurance seems like a must anyway these days, given all the risks surrounding travel (cancellations, border closures, etc.).