This is going to sound extremely stupid and I have tried avoiding mentioning it because I know this might not go over well, but... the thing I want is a pet gryphon.
I know that sounds like a joke and maybe a little insane, but it is important to me. Pretty much I have been having dreams about the same gryphon my entire life and I have grown a level of affection for it. I am not obsessed with it, yet at the same time I want it to be real and to be my pet. Early on in my journey into Christianity I realized that it was only possible through God because I knew/know he will make a new earth and saw this as a possibility that he will create it.
Sorry all if this sounds stupid.
I doubt anyone can give you a specific Biblical reference to Gryphons in Heaven, but with that caveat ... your chances are much better in heaven than here on earth (look at the creatures that surround the throne [Ezekiel 1:4-14; Revelation 4:5-8] and a Gryphon is less "strange" than them) and there is at least Biblical support in heaven for all creatures encountered there to be FRIENDLY. [Isaiah 11:1-10]
So your chances are at least "fair" (it is not an unreasonable impossibility) in my humble opinion.
Just a passing thought, but if your dream is of an "angelic being", then it is more a "guardian" and "friend" than a "pet" ... but I am not Joseph or Daniel and INTERPRETING DREAMS is not my thing. However offering advice from scripture IS MY THING, so "TEST THE SPIRITS" [1 John 4:1-6] and "HOLD ON TO WHAT IS GOOD" [1 Thessalonians 5:19-22].
Reference Verses:
[Eze 1:4-14 NIV]
4 I looked, and I saw a windstorm coming out of the north--an immense cloud with flashing lightning and surrounded by brilliant light. The center of the fire looked like glowing metal, 5 and in the fire was what looked like four living creatures. In appearance their form was human, 6 but each of them had four faces and four wings. 7 Their legs were straight; their feet were like those of a calf and gleamed like burnished bronze. 8 Under their wings on their four sides they had human hands. All four of them had faces and wings, 9 and the wings of one touched the wings of another. Each one went straight ahead; they did not turn as they moved. 10 Their faces looked like this: Each of the four had the face of a human being, and on the right side each had the face of a lion, and on the left the face of an ox; each also had the face of an eagle. 11 Such were their faces. They each had two wings spreading out upward, each wing touching that of the creature on either side; and each had two other wings covering its body. 12 Each one went straight ahead. Wherever the spirit would go, they would go, without turning as they went. 13 The appearance of the living creatures was like burning coals of fire or like torches. Fire moved back and forth among the creatures; it was bright, and lightning flashed out of it. 14 The creatures sped back and forth like flashes of lightning.
[Rev 4:5-8 NIV]
5 From the throne came flashes of lightning, rumblings and peals of thunder. In front of the throne, seven lamps were blazing. These are the seven spirits of God. 6 Also in front of the throne there was what looked like a sea of glass, clear as crystal. In the center, around the throne, were four living creatures, and they were covered with eyes, in front and in back. 7 The first living creature was like a lion, the second was like an ox, the third had a face like a man, the fourth was like a flying eagle. 8 Each of the four living creatures had six wings and was covered with eyes all around, even under its wings. Day and night they never stop saying: " 'Holy, holy, holy is the Lord God Almighty,' who was, and is, and is to come."
[Isa 11:1-10 NIV]
1 A shoot will come up from the stump of Jesse; from his roots a Branch will bear fruit. 2 The Spirit of the LORD will rest on him--the Spirit of wisdom and of understanding, the Spirit of counsel and of might, the Spirit of the knowledge and fear of the LORD-- 3 and he will delight in the fear of the LORD. He will not judge by what he sees with his eyes, or decide by what he hears with his ears; 4 but with righteousness he will judge the needy, with justice he will give decisions for the poor of the earth. He will strike the earth with the rod of his mouth; with the breath of his lips he will slay the wicked. 5 Righteousness will be his belt and faithfulness the sash around his waist. 6 The wolf will live with the lamb, the leopard will lie down with the goat, the calf and the lion and the yearling together; and a little child will lead them. 7 The cow will feed with the bear, their young will lie down together, and the lion will eat straw like the ox. 8 The infant will play near the cobra's den, and the young child will put its hand into the viper's nest. 9 They will neither harm nor destroy on all my holy mountain, for the earth will be filled with the knowledge of the LORD as the waters cover the sea. 10 In that day the Root of Jesse will stand as a banner for the peoples; the nations will rally to him, and his resting place will be glorious.
[1Jo 4:1-6 NIV]
1 Dear friends, do not believe every spirit, but test the spirits to see whether they are from God, because many false prophets have gone out into the world. 2 This is how you can recognize the Spirit of God: Every spirit that acknowledges that Jesus Christ has come in the flesh is from God, 3 but every spirit that does not acknowledge Jesus is not from God. This is the spirit of the antichrist, which you have heard is coming and even now is already in the world. 4 You, dear children, are from God and have overcome them, because the one who is in you is greater than the one who is in the world. 5 They are from the world and therefore speak from the viewpoint of the world, and the world listens to them. 6 We are from God, and whoever knows God listens to us; but whoever is not from God does not listen to us. This is how we recognize the Spirit of truth and the spirit of falsehood.
[1Th 5:19-22 NIV]
19 Do not quench the Spirit. 20 Do not treat prophecies with contempt 21 but test them all; hold on to what is good, 22 reject every kind of evil.