This is a really interesting question that led me into some new realizations that I had not previously made, so thank you for asking it!

In short, yes, Bahrain and Qatar share a deep history. Historically, Bahrain was the dominant of the two and the Al Khalifah, who rule Bahrain to this day, were effectively rulers of Qatar from 1766-1868 until the Al Thani, the present rulers of Qatar, led a revolt 1867-68 which was concluded with a British treaty that de-facto recognized the Al Thani as the rulers of Qatar. In 1895, with Qatar then effectively under Turkish protection, two Qatari sheikhs attempted to invade Bahrain. The territorial disputes and mutual claims to various parts of each other's territory remained unresolved long-after independence, and their dispute over control of the Hawar Islands was only resolved by the International Court of Justice in 2001.

The question of the design of their flags is linked to the protection and enforced truce agreements the Britain signed with Bahrain, Qatar, and the seven Emirates that today make up the United Arab Emirates (then-known as the "Trucial States" because of these truces.) The 1820 general maritime treaty signed with six of these emirates (Fujairah did not yet exist) and eventually with Bahrain.

The "maritime" nature of the treaty was that many if not all of these entities were engaged in intermittent piracy, particularly the Al Qasimi who rule over both Sharjah and Ras al-Khaimah. The current ruler of Sharjah, Sultan Muhammad Al Qasimi, who has a PhD in history from Exeter University, disputes this in works such as "The Myth of Arab Piracy in the Gulf." I haven't read his book or evaluated his claims, but I'm not familiar with any other historians who share his view.

Among the terms included in the treaty was that all friendly Arab states should fly flags of red borded by white, referred to in the treaty as the "white-pierced-red" flag. The default flag color for virtually the entire region being otherwise just a plain red.

The result is that all seven of the UAE's constituent emirates have flags of red and white: https://en.wikipedia.org/wiki/Emirates_of_the_United_Arab_Emirates

The flags of Ajman and Dubai, in particular, are nearly identical to that of Bahrain except for the serrated edging.

This serrated edging on Bahrain's flag was added in 1932, at the same time that Bahrain adopted a coat of arms that also includes the serrated edging (dancetty, in heraldic terms.) The number of serrations has greatly been reduced over time, from 32 between 1932 and 1972 down to just five today.

One key piece of information that I have struggled to find in answering this is when/how/why did Qatar adopt and alter its flag. The official answer from the Qatari government points to the flag and the serrations also being adopted in 1932., but there are enough nonsensical/propaganda-ish points made on that page that I'm not entirely prepared to accept it at face value. For one thing, every other non-Qatari source I have been able to find on the question has insisted that the Qatari flag was formerly red like the others and that "Qatari Maroon" was adopted in 1949.

So to sum up: yes, these countries have a long shared history between themselves and with other countries in the region, there are some historical reasons relating to various treaties for why white and red flags predominate, but some of the specifics the timeline and decision making regarding the adoption of specific characteristics of these flags doesn't appear to be entirely clear from the sources that I have looked into, particularly as several of them are contradictory. A firmer answer than the above I suspect would require research into the diaries and records of the British political agents who largely made these decisions, particularly Charles Belgrave, who was the British Political Agent responsible for both Bahrain and Qatar in 1932.

While his diaries have been published, I unfortunately do not have access to them.