I am trying to understand why both 弟 and 第 kanji have the same 音読み pronunciation ダイ/テイ? The common element 弔 is a variant of 吊 - is its Chinese pronunciation close to ダイ/テイ?
2 Answers
So 弟 is actually a 象形{しょうけい} (pictograph) and it is pronounced ダイ (among others of course).
On the other hand 第 is a 形声{けいせい} (kanji in which one element suggests the meaning and one the sound) and from the link you can see that the pronunciation comes just from 弟. So I guess you should really look at 弟 as the phonetic part rather than 弔.

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They have the same on'yomi because「弟」and「第」were originally the same character, representing a word to do with sequence or order; younger brother is an extension of this meaning (via ordering of male siblings).「第」was created from「弟」by corrupting「丷」to「⺮」. In Mandarin Chinese, the two characters also sound identical (dì), and bears distant similarity to ダイ/テイ.
Despite「弔」looking like it shows up in「弟」and「第」, they are unrelated.
For reference,「弔」depicts a person「人」wielding a bird-capturing device; the device is comprised of an arrow attached to a very long rope, and is the original character of「/繳」.
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「弟」is comprised of a handle of a pole-arm weapon「必」and a depiction of binding rope inscribed in an「S」shape. It originally represented an ordered bundle of weapon handles, extended to mean order, then ordered male siblings > younger brother.
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「必」is not related to「心」!「必」was originally「戈」(dagger-axe) with the blade part omitted, leaving the handle part. The modern meaning certain is a phonetic loan.
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合集33208The handle part of the weapon is now written as「柲」. Later on, decorative marks were added to the character, leading to the modern form.
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Compare「心」, which depicts a heart with the heart chambers becoming exaggerated, then later heavily abbreviated.
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The「丷」part of「弟」was an eventual detachment from the main body of the handle. In some variants,「丷」was corrupted into「艹」, then again corrupted into「⺮」, leading on to the modern form of「第」, which preserves the meaning order or sequence and is not used for the meaning younger brother.