Is there any standard size in Custom PCB Manufacturing?

Sometimes PCB makers have a per-panel price. They will tell you the size they are using..or more importantly, what area you have to work with-- usually usable area (not panel size) is at least 10" x 16", maybe up to 10.5" x 16.5" or a bit more. That would be a small panel. For large production I believe larger panels are more typical. Typically phenolic single sided boards use much larger panel sizes. This reference lists typical standard PCB panel sizes as follows:

Panel borders (maybe 0.5" on each side) take away from that area.

It is not particularly important unless your PCBs (or your self-panelized PCBs) are getting up to the size where you only get a few from a panel- materials are not really the biggest part of the PCB cost in prototype volumes despite you being charged per square cm or inch. If your PCB or panel is approaching or exceeding about a letter-size or A4 sheet and your quantities are more than hundreds (of that size, it could be thousands of panelized boards) it might be worth worrying about.

When you order loose boards you don't have to pay for router kerf and tooling strips either as PCB area (well, you do but it's not figured directly into the costs) or as shipping mass, but extra costs are incurred later.

It might be interesting to play with the online calculators that many PCB makers have and see if panelization figures into their calculations. You would expect the cost per square inch to not follow a straight line as the size gets bigger. Keep in mind though that getting large defect-free areas gets more expensive as the area increases so a large panel (without X-outs permitted) may be more expensive per square inch than a smaller panel. It may not be important if the technology level is low, but for multilayer boards with fine pitch and tiny holes it is not negligible. Often the online quotes are not the final quote in such cases, and they will come back with an adjusted quote (hint: it is never lower).

What's important to the fabrication shop is not the size of the individual board, but the size of the panel it will be made in. Meaning, typically PCBs are made with multiple images on a panel, and then cut apart either before delivery or after component assembly.

And, yes, there are (semi-)standard panel sizes.

Among those standards, different shops will be set up for different ones. However, every shop I know offers 18 x 24 inches as a standard size.

There are lots of other sizes available, so if 18 x 24 doesn't work for you, contact your shop to see what else they offer.

If the total area of your order (plus some allowances I'll mention below) is less than the smallest panel your vendor can handle, you'll end up paying for some waste. If you try to absolutely maximize the usage of a panel, the vendor is likely to run one extra panel to allow them to still deliver your order if there's a defect in one location, and again you'll pay (a fairly small amount) for this extra panel to be made and then discarded. If you design your board to fit (for example) 12 to a panel, and then only order 10 boards, you're likely to get about the best pricing you can hope for.

When deciding how to panelize your board, you also need to consider the routing allowance needed between the boards and gutter allowance along the edges to allow for handling. In my experience, typical minimums for these are 0.2 and 0.25 inches, but different shops may have different requirements, so confirm this with your shop.

If the boards will be assembled (pick & place'd) in panel form, you should also verify the edge gutter requirement with your assembly shop.

to not incur extra cost for the loss of the material PCB Manufacturer faces

I think this is your key assumption which I would challenge. While it is great that you care about other people and nature not to waste the materials, I am afraid that it is simply out of your control in this case.

Each square inch of the board is a cost for PCB house. They strive to minimize losses, they combine boards on the sheets, rotate them, use other tricks.

Since PCB manufacturing is nothing than just a business, what you primary would care about is your - end customer - price for their end product - the PCB. And of course less board size should (usually) lead to less variable cost (the cost which really depends on the production of your board - for example, FR4 material sheet size, amount of mask, etc).

In general, if you are still curious beyond the price as indication of the manufacturer's usage of materials (and their quality :) you always can ask them which equipment they use, how they use it, what they do with your Gerber files etc, but I bet they will try not to disclose much of this info because usually this information is not for your eyes as a customer.

Update: as @BeB00 summarized in the comment

so the correct answer is to just get quotes for different sizes and see

and I would add to it that you must design your board so that device would work as expected, can be mounted into place, and be serviceable (if possible). Trying to stick to some size without real product requirement, from my point of view, is nonsense.

Respectable PCB houses usually advise their clients on the quality of their Gerber designs, and even tell clients if there're any elements which can be changed to (a) decrease production cost/minimize productions issues, and thus (b) decrease total cost for customer.