Why are they not asking me about my passport number for my booking?

This is absolutely normal. A passport is not required to purchase a ticket and you may be doing it for someone else or someone else could be doing it for you, so only need the full name of passenger is needed. This makes the process of buying quicker.

Any time after the booking goes through, passengers (which may not be the purchaser) can complete their information, up to right at the moment of booking, although some jurisdictions require that info is send a certain amount of time ahead of travel.

A passport is required for international travel but not always for domestic travel. I am not sure there are even exceptions to the international travel requires a passport common knowledge (they may be some close countries that allow travel between them without a passport, a good other question!).

Not asking for passport info at booking time, even when required for travel, can reduce issues since someone may renew or lose a passport in between booking and travel. In both cases, most countries now change passport numbers. At least the expiry date will change.

It very much depends from where to where you travel. If I fly from Europe to the USA, I have to supply my passport number when booking my ticket. OTOH, if I book a ticket to another Schengen country, even with the same airline company, I don't have to supply a passport number. I won't go through immigration when travelling between Schengen countries, and airline companies are not asked for passport numbers. Hence, they don't ask (and I would not be surprised if they are not even allowed to ask for the passport number for such flights).

If you are not crossing international boundaries, then passport information is not required.

It also may not be required if you do not need a passport for the itinerary (for example, flying within the Schengen zone, GCC citizens flying within the GCC, etc.)

In my experience KLM, Emirates, Qatar and Etihad all allow customers to update their passport information online. They even notify you when your passport has expired to update that information.

However, doing so is only required if you plan on checking in online.

Otherwise, if required, your passport information is collected at the check-in desk or the self-service check-in kiosks.

Likely you will get an e-mail or (if asked to be contacted in an other way) a text message or a letter, to add your passport information to the booking.
For many travels, you do not need to add the information before you check in. If that is so for your travel, you will be asked to add it as part of your checking-in.

There are travels, even international ones, where you do not need to give a passport number. If so, you do not need to add your passport number at all.

If you do not want to wait, you can usually go to the website of the airline, to your travel agent if they booked for you or call the airline company on their help line. Once you logged into your account or called up your flight with the information given to you, you can find the information where and how to add your passport number.

Where I have written 'passport' it can also be an other accepted travel document if that is acceptable for your flight.

It is possible that your airline does not allow passengers to alter or add passport numbers after booking the flight, in that case you may have to contact the airline to have an agent to add it for you. (Although this is less likely with airlines that do not ask for the document details at the time of booking.)

In some cases, the information can be added as late as when you are in the airport to check in just before the flight, but better ask earlier, as some international travel can only be done if the information is at the airline at least 24 hours (or sometimes more) before travel.

It's very simple. A Passport number (or other valid travel doc) is required to travel, not to buy a ticket.

It's not uncommon for first time travelers to buy tickets while their Passport application is still being processed.

Note, if you do not or cannot provide a valid Passport, you will be denied boarding with no compensation.

You can modify the Passport number any time, probably without cost, either at the airline's website or by calling reservations.