Kicked out of hotel in morning when booking through booking agency (asked to pay for extra day)

This is perfectly normal. Checkout times at hotels are usually between 10:00 and 12:00 and you need to pay something extra (sometimes for the next night, sometimes less - depending on hotel rules) for overstaying.

Most hotels offer their guests to leave their luggage for free for the rest of the day. So in this case I would check-out of the hotel in the morning, leave my luggage with the reception and then go out to enjoy the rest of the day.

Short answer, to echo the other answers: it's entirely legal. Checkout time is checkout time, typically anywhere from 9am - 1pm (0900 - 1300). Depending on the situation, you may be able to extend this to within an hour or so of check-in time (typically 3pm), and a late checkout may or may be free. Factors which affect the situation include: what the arrival situation looks like for the next night (if it's a hotel that does a good business with weddings, a late checkout on a Saturday will generally be a minor miracle: wedding groups mean lots of early check-ins or at least attempts at such), membership level in the hotel's loyalty/rewards program, how much of a regular history you have with the hotel (4 one-night stays (especially if regularly spaced, e.g. every three months, as might be the case for a traveling salesman) generally count more for this than a single 14 night stay: the hotel will figure that you'll be coming back), the ratio of guests who have checked out already to guests with late checkouts, and how you made your reservation (in approximate order: directly with the hotel, through the hotel chain's website, through a credit card issuer's travel agency, and then basically everything else). If all else fails, a large tip for the front desk agent may get you a late checkout (in my earlier life behind a hotel front desk, for my rare morning shifts (I mostly worked night audit), I could most days get away with a total of 2 hours worth of discretionary late checkouts without getting yelled at by management or the housekeepers; a "$100 handshake" could reliably get one an extra hour).

None of this is any good for your situation: the best case for a late checkout is 3pm, hours less than you need. All is not lost, though. Depending on the hotel, there's a thing called "day use" or "zero night stay". If you happen to choose a hotel where the hotel figures they're able to flip the room and sell it again for the night, this may not be that expensive. The staffing practices of the hotel and their demand pattern are what govern day use availability. Generally, the more full-service the hotel is, the more likely they will have someone around in the evening to flip a room. If they're close to an airport or in the downtown of a major city, they're likely to be able to have someone walk in later wanting a room. On the other hand, limited service properties (e.g. motels), tend to have no one working at night besides the front-desk agent, so day use isn't an option (most hotel software will not allow a reservation for day use, by the way: generally the only way to do day use is to walk in or do it as an extension of the stay).

Beyond day use, most hotels will be willing to store your luggage etc. for a few hours, though it's best to bring up the possibility when you check-in. Factors which affect this are similar to those affecting whether you can get a late checkout, but you have a far greater chance of success.

TL;DR; Hotel was right (I was wrong) - to pay for one extra day was most appropriate solution.

As an addition to the above answers there is a long discussion in comments around was it legal to keep my belongings (this is what I was looking for).

In one of comments @piet.t refers to the hotelkeepers liens article.

Hotelkeeper’s liens allow a hotelkeeper to hold personal property that a guest brought with them into the hotel, as security for payment. Such liens are also known as “innkeeper’s liens”. Hotelkeeper’s liens usually apply only to the “baggage” of a person, and not their automobiles.

In Spanish (Canary Islands belong to Spain) it is called gravamen de hotelero

So it was perfectly legal to keep my belongings until I pay for the loss of the room for the day.

My personal advice - do not use online websites to book your holiday, go to travel agency in person, additional cost is usually worth the issues avoided. If something goes wrong I am sure they will be more helpful than phone operator (if they don't they know that day after holiday ends you will be walking through their office door and talking to their manager ;) )

There is one thing missing from the other answers:

In every hotel I ever stayed in (in total about 50), they tell you in person what the check out time is while you check in. They tell you even if you don't ask. If you book through a website or a travel agency doesn't matter, the one telling you about checkout times is the person who checks you in at the hotel. If they tell you something and you don't understand what they mean, ask them what they mean, because it might be important.

In cheaper hotels you can usually check out up to 1 hour later than the checkout time without any problems, but the more expensive the hotel the more likely there will be an extra charge.

If you realize that the stated checkout time won't work for you, you need to let the hotel know in advance. They will always be able to find a reasonable solution, be it a late checkout, storing your luggage for you, or something else - they might even allow you to keep the room at no charge, as long as you're gone the next morning. But the prerequisite is that you tell them first. By occupying a room without booking it, you are one of the bad guests, and they have no reason to be nice to you.