Ok, probably this tunnel is not so secret after all.
In fact, I was more than 100% sure that the name of this place is Kolga Tbilisi Photo Tunnel, as I have seen a photo exhibition at this tunnel, but Kolga.ge is stubbornly silent on this matter.
Turns out the name for this place is "Leghvtakhevi Tunnel" and the owner of the place is Meidan Group. (see comments)
Kolga Tbilisi Photo organises exhibitions throughout the city, but it doesn't mean that Kolga Tbilisi Photo owns the tunnel, Baratashvili Bridge, galleries and etc.
Located on one level with the river flowing through the Sulfur bath district
Tbilisi sulfur bath district
Abanotubani is one of the most attractive neighborhoods in Tbilisi. It's famous for its public sulfur baths, and very spectacular views.
I believe Tbilisi Sulphur bath district is one of the most iconic Tbilisi sights. Personally, I doubt there are many tourists who have not seen Abanotubani (Tbilisi bath district) while visiting Georgian capital
Kolga Tbilisi Photo exhibition at Leghvtakhevi Tunnel
Photo exhibit in the tunnel
The tunnel is quite interesting and a real hidden gem of Tbilisi. Bypassing tunnel you can reach a small pier with a great view towards Metekhi church.
View towards Metekhi church