Not really. Triangles were how state highways were signed until 1983 when they added many miles of routing and divided the system into state primary and state secondary. The secondary highways continued to use the triangle while the primary highways got a new shield which the TN 153 shield in your photo has.
They still sign with the different shields even though in recent years they ditched the primary/secondary delineation in their state highway system.
Some Tennessee counties use the blue pentagon for their county routes.
Their definition of what constituted a primary highway was that it “primarily connects cities”. So essentially the major routes or shortest routes between county seats or other relevant cities were primary, and everything else was deemed to be secondary. At times you would have routes that flipped back and forth between primary and secondary. TN 46 is a good example of this, as it is secondary from its beginning at US 79 to Dickson (including across the Cumberland City Ferry), then it’s primary from Dickson to the intersection with TN 100 and TN 7 in Bon Aqua, and then it’s secondary again from TN 100 through Leiper’s Fork to its ending at US 431 in the Grassland community north of Franklin.
And I am aware of only three counties that use the blue pentagon for their numbered county routes, and they are all in East Tennessee: Meigs, McMinn, and Monroe.
Also, if you look at various roads in rural Tennessee in Google Maps and you see a four digit designation on a rural road, either in a circle or a rectangle, and it’s not in one of the three counties I mentioned, I guarantee you it isn’t signed. It is most likely a state-aid designation that was given by the state for roads that they help with maintenance costs, such as bridges. For example, I used to live off of a roadway in Cheatham County that Google Maps shows as being designated “1948” in an oval. As far as I know, it has never been signed as such, and has always been referred to as Kingston Springs Rd. But it does have two bridges, one over a major CSX rail line and another over a large creek, that likely receives maintenance/inspections by TDOT, and it does receive salt treatments and plowing during snowy weather events, most likely also by TDOT.