Imagine a primordial black hole so heavy that it is a blackbody and there are many planets orbiting around it, there is a moon thriving with intelligent life orbiting a gas giant. I am wondering would the intelligent alien species be able to tell a blackhole "Sun" against a fusion furnace?
Mismatching Spectrum and size
If their "sun" emits the same total energy of light as our Sol, then it will be only:
total luminosity : matches Sol = 5e26 watt
diameter: 1/ 10 000 th of a proton. (1e-24 m diameter)
peak emission: about 4.2e16 electonvolt. That is deep, deep, deeeeeeep cosmic ray, at about 6 millijoule PER PHOTON.
It would also only mass about 1 tonne,
and they would have a few microseconds to observe it before it evaporates.
A black hole large enough the be the gravitational center of a solar system, especially a "black hole so heavy that it is a blackbody" will have virtually nil detectable radiation of its own. Maybe from an infalling accretion disk, but that has a very visible and distinct signature.
Here' a nice tool to play around with, to see the size & mass & luminosity of black holes.
It's really hard to have a black hole involved in a habitable system. As others have pointed out, the black hole you describe would be a tiny point of light, and it would immediately evaporate in a burst of gamma radiation.
Another alternative would be a quasi-star, a massive object heated by a black hole accreting matter at its core. These would be obviously unlike normal stars, being thousands of times as massive as the sun and outputting as much light as a small galaxy. They might provide heat and light to a planet orbiting another star that itself is too dim. However, they would only last 7 million years or so, and since stars move, it's unlikely a planet would end up in the habitable zone for even that long.
A stellar black hole with an accretion disk seems more plausible, though you have the problem of explaining how the planet survived the creation of the black hole in suitable condition for life, and the accretion disk seems likely to produce harmful radiation and to be too unstable a source of heat and light for life to form.
However you arrange it, it seems unlikely for complex life to evolve locally in the time available: the aliens would almost have to themselves be alien to the world they're on, having come from somewhere else. Refugees from the formation of the quasi-star, settling a world that it's made habitable for the next few million years?
Assuming that they follow the same path of scientific evolution that we had, they would notice that its mass is rather large.
We start by determining the mass of the Earth. Because we know the radius of the Earth, we can use the Law of Universal Gravitation to calculate the mass of the Earth in terms of the gravitational force on an object (its weight) at the Earth's surface, using the radius of the Earth as the distance.
Knowing the mass and radius of the Earth and the distance of the Earth from the sun, we can calculate the mass of the sun (right), again by using the law of universal gravitation. The gravitational attraction between the Earth and the sun is G times the sun's mass times the Earth's mass, divided by the distance between the Earth and the sun squared. This attraction must be equal to the centripetal force needed to keep the earth in its (almost circular) orbit around the sun. The centripetal force is the Earth's mass times the square of its speed divided by its distance from the sun. By astronomically determining the distance to the sun, we can calculate the earth's speed around the sun and hence the sun's mass.
A similar approach would lead to estimating the mass of the light emitter.
However be advised: a black hole with peak emission at 400 nm would have a radius of 10 nm, a mass about 1 millionth of Earth and a luminosity of 1 millionth of a Watt. In this world it's probably true that the Sun orbits around the Earth, but it would be very feeble.