Yes, it’s seen as more formal, which is by nature more respectful. In other languages, you sometimes have formalized speech as an integral part of the language, while in English we don’t really have that, so I think that is why many prefer to revert to the antiquated speech as a substitute.
When many people pray, they revert to the 16th century English language as it is considered a respectful language. (hiho hiho post 6 to Ruy Dias de Bivar)
Often a person offering prayers supposes that the pronouns "Thee, Thou, Thy, Thine" infer a closer or more intimate address toward The God, or that the Bible language form demonstrates a greater religiousness or respectfulness. But that was not the intent of the translators (though it may well be the desire of the one praying). They desired to present the information in the grammatic sense of the original writer.
So, for instance, why do we see the "Thees and Thys" in the KJV? It is because one of the great gifts of the KJV translators was to insist on a correct English grammar, so that in the translation one may know whether the subject referred to is second person singular or plural.
When the modern versions use the pronoun "you" for both singular and plural, or if the distinction is not carefully made using the KJV, often a wrong meaning is read into the particular verse, leading to misinterpretation and wrong doctrine.
Any person using the KJV knowledgeably can avoid falling into that trap, by knowing whether the application is meant for an individual (thee), or whether it is meant for, say, the church (you). But without correct interpretation, a modern version leaves the reader at sea as how to apply Biblical instruction.
(Also, in addressing The Father God, the prayer will more fitly address Him as "Thou" rather than as "You." This would be merely good grammar, not to be thought of as more saintly or reverent or religious or more respectful.)
Just a thought ...