The significance of those pledges in bound up with the whole story of Onan violating his obligations with leverate marriage to Tamar, as recorded in Genesis chapter 38. Assuming this is understood, why would those items be significant?
Tamar's first husband was struck dead by God for his wickedness, then the brother, Onan, married her. But God took Onan's life, for violating the law of leverate marriage. Their father Judah then resorted to trickery due to him not intending to give her his third son once he was old enough (for fear God would kill that son too - Genesis 38:11.)
This resulted in Tamar going into purdah (living back at her father's home), being told to wait until Judah's son Shelah would be old enough to marry her. But Tamar was no fool. When Judah's wife died some years later, after a period of mourning, Judah traveled to the area where Tamar was living. She took off her widow's garments and veiled herself as did shrine prostitutes, to sit by the roadside. He noticed, slept with her (not realising she was his daughter-in-law), thus making her pregnant, and so she raised up twins for her dead first husband. She tricked Judah who had tricked her, in order to uphold the law of leverate marriage that he had had no intention of keeping.
When confronted with all this, Judah admitted, "She is more righteous than I am because I didn't arrange for her to marry my son Shelah." (Gen. 38:26)
Now, the pledges. Tamar required paying, up front, for her sexual favours. Judah apparently did not have cash in hand, saying he would send her a goat as payment. Not good enough. Tamar required some personal items, to keep until the goat arrived when she would be required to return the pledges, but she planned to disappear before then. The arrangement secured Judah's pledges, which she had no intention of returning until her pregnancy would be noticed. His identification seal was almost like a set of fingerprints, as proof of identity. There was likely a cord attached to this seal to enable it to be worn around the neck. Then there was his walking-stick (or staff).
They were significant because Tamar's risky plan depended on her getting pregnant by Judah, then exposing him as the father. The Judaic law required two or more 'witnesses' to establish such legal matters:
Deuteronomy 19:15: "One witness shall not rise up against a man for
any iniquity, or for any sin, in any sin that he sinneth: at the mouth
of two witnesses, or at the mouth of three witnesses, shall the matter
be established."
The seal (with attached, intact cord, to prove it hadn't dropped off his neck accidentally and had been picked up by anybody) was one 'witness'. The second one was Judah's personal staff. That was not the kind of item he would drop or lose without noticing, and without retrieving. It likely had personal marking to identify it as uniquely belonging to Judah. Tamar's plan required not getting the goat so as to keep the pledges, thus when she confronted Judah in a state of pregnancy, she could prove he was the father before he got her killed for adultery. As good as a DNA test, for those people. Even if he had denied it, the two silent witnesses identified him.
"The seal and its cord" is the wording in the NIV and the NLT. Footnotes read,
"Probably a small cylinder seal of the type used to sign clay
documents by rolling the seal over the clay. The owner wore it around
his neck on a cord threaded through a hole drilled lengthwise through
it."
So, the significance of what really were two pledges lay in the law of at least two witnesses for such serious accusations. But there could be more, given the seal was used for legal documents, and a staff (or walking stick) was a symbol of authority. The father of Tamar's twins had to legally own them as their father, and his authority ensured they were treated as superior in inheritance rights to even his own (third) son, Shelah. Further, when the first twin showed his hand, a midwife put a scarlet cord around his wrist, but that was not the first-born twin, who was called Perez (meaning 'breaking out'). The other was called Zerah (meaning 'scarlet' or 'brightness'). Perhaps the cord attached to Judah's seal was scarlet? Significantly, Perez became the head of the leading clan in Judah, and the ancestor of king David (Ruth 4:18-22) and ultimately of Christ (Matthew 1:1-6).
The staff of Judah is significant in Genesis 49 for it foretells in prophecy:
"The sceptre shall not depart from Judah, nor a lawgiver [the ruler's
staff, NIV] from between his feet, until Shiloh come; and unto him
shall the gathering of the people be." Genesis 49:10
Great significance, all over the place, in that Genesis account!