Is believing that humans have the same right to exist as other species inherently the same thing as?

Apparently they have more rights.

   In nature there are natural forces that keep animal populations in check.  Man has overcome those forces mostly by science.  But nature will win in the end. 

   Either we curb our population or nature will. The later it is the worse it will be. 

simple rules we have a right to live and the same right to die , we kill animals and clear forests for the land for food and cut down trees ,and destroy and make more rubbish , can any other animal be as pollutive as humans ..a lion kills and eats everything , humans leave part of the animal for rubbish .

All living creatures strive to exist. Nature exists in a balanced and integrated manner. All animals live according to their instincts except for humanity. They alone ignore Nature's laws. As a result of our destructive and unsustainable tendencies, we are depleting the resources and animal life we need for a good future. 

Nature is trying to wake us up to the consequences of our disregard of her laws. Covid has been the first global blow to make us stop and think of our actions and the need for us to change, as a collective.

We must connect above our differences in order to live in balance with each other, which in turn will have a positive impact on our environment. When we care for others, we won't want to pollute and poison other people. We won't want to approve policies that impact negatively on people's livelihood, which sends them spiraling into financial ruin. 

So many decisions are made without consideration for environmental or human impact.

We need to change. Our comfortable future depends on in.

NO. If humans have the same rights as other species, we will try to prevent them from getting too numerous.

Even with current per-capita consumption and population we will wipe out other species for decades to come

No. Google Deer + the Kaibab Plateau to see what happens when population gets out of check. Remove the checks on populations, as by removing predators, and populations soar past the capacity of the environment to support the population. It might be starvation or it could be a disease that ravages the population. Who said COVID-19 was all bad? Indirectly, it could be the factor that corrects a political situation as well as overpopulation by fools.

Apparently they have more rights.

   In nature there are natural forces that keep animal populations in check.  Man has overcome those forces mostly by science.  But nature will win in the end. 

   Either we curb our population or nature will. The later it is the worse it will be.