Does Buddhism claim to be only way to liberation?

I hold the view that Buddhism may not be suitable for all individuals.

I think the suttas say that some people are quick to learn and benefit, and others less so.

And that Mahayana doctrine adds, that even for example an animal may benefit from hearing the Dharma, if not in this life then perhaps a next.

In my opinion, this presents a limitation of Buddhism compared to other religions, where prayer may be an easier means to gain insight.

That's not obvious to me, but that would require discussing the nature of prayer in other religions, so not really on-topic.

In one sutta, the Buddha mentions the need to be fortunate to encounter his teachings.

I read a slightly different message in that kind of sutta, i.e. that, "Now that you have been so very fortunate as to encounter Buddhism, do not waste this rare opportunity."

Doesn't this exclusivity make Buddhism an exclusive religion for attaining liberation?

I wouldn't say exclusive -- it begins like this in SN 6.1:

Then the Blessed One, having understood Brahma's invitation, out of compassion for beings, surveyed the world with the eye of an Awakened One. As he did so, he saw beings with little dust in their eyes and those with much, those with keen faculties and those with dull, those with good attributes and those with bad, those easy to teach and those hard...

Having seen this, he answered Brahma Sahampati in verse:

Open are the doors to the Deathless
to those with ears.
...

"To those with ears" isn't very exclusive -- no more exclusive than Christianity, IMO ("He who has ears to hear, let him hear!" in Matthew 11:15).

And the Dhamma is famously "inviting inspection":

The Dhamma is well declared by the Bhagavā: visible here and now, immediate, inviting to come and see, effective, to be individually ascertained by the wise.

And there are many forms of Buddhism, including some with practices which may be similar to what you call "prayer".

There are people that claim things about Buddhism, (claim is used here as a verb) but a Buddha doesn't force teachings upon another.

If you think about the diversity of the pali canon teachings, (I think around 84,000) it's more reasonable to say that Buddhism isn't a particular teaching, or a specific thing that holds a tangible one-sided belief about the human experience, but more a calling to discover what form of inquiry works for you. It has an application for many people from all walks of life, from the homeless wonderers to the rich and powerful, and all the people in-between.

However, the short answer is no, the Buddhist context is not the only way. You can crack this nut using other traditions or methods, secular or non-secular.

yes, It clearly mentioned in suttas!

one may think that it's kind of a overestimating the Buddhism, But in-fact it's an obvious and profound fact!

if one can understand the real Buddhism, he or she really can experience the trueness of the dhamma!

and it's like if one can go to a topmost place in a ground, he can observe the surroundings well and he knows that all other points a situated lower than his standing point. similarly, those who know at least the first glimpse of true dhamma knows as to what exactly the complete freedom is {he sees the way leading to it}

The first set of verses below say that there's no noble (ariya) contemplative (i.e. the four types of enlightened persons) outside the Buddha's religion (i.e. dhamma-vinaya).

The second set of verses say that there is no better path than the Noble Eightfold Path, and there's no higher truth than the Four Noble Truths. This is the only path to the purification of insight.

Also, there's no way to end suffering without realizing the Three Marks of Existence.

That said, it's theoretically possible for followers of other religions to attain the complete end of suffering, as long as they realize the Four Noble Truths, Three Marks of Existence and practice the Noble Eightfold Path.

There is no track in the sky, and no recluse outside (the Buddha’s dispensation). Mankind delights in worldliness, but the Buddhas are free from worldliness.

There is no track in the sky, and no recluse outside (the Buddha’s dispensation). There are no conditioned things that are eternal, and no instability in the Buddhas.
Dhammapada 254-255

Of all the paths the Eightfold Path is the best; of all the truths the Four Noble Truths are the best; of all things passionlessness is the best: of men the Seeing One (the Buddha) is the best.

This is the only path; there is none other for the purification of insight. Tread this path, and you will bewilder Mara.

Walking upon this path you will make an end of suffering. Having discovered how to pull out the thorn of lust, I make known the path.

You yourselves must strive; the Buddhas only point the way. Those meditative ones who tread the path are released from the bonds of Mara.

“All conditioned things are impermanent”—when one sees this with wisdom, one turns away from suffering. This is the path to purification.

“All conditioned things are unsatisfactory”—when one sees this with wisdom, one turns away from suffering. This is the path to purification.

“All things are not-self”—when one sees this with wisdom, one turns away from suffering. This is the path to purification.
Dhammapada 273-279

It's important to make a distinction between the understanding and the vehicle. Buddhism is clearly going to tell us that the only way to reach liberation is by grasping the nature of dukkha and tanhā: seeing how we create our own idiosyncratic inner world, and how misery arises when that inner world grinds up against reality. Buddhism is also going to tell us that the best vehicle for attaining that understanding is one of the Buddhist paths, but it won't say that in such strident, exclusive tones. In fact, Buddhism has a history of combining with and integrating other religions — Daoism, Shinto, Bonn, even Hinduism, and Christianity, and secularism — to produce 'hybrids'. There's nothing stopping someone in a different faith from gaining the understanding.

From the Buddhist perspective you choose a vehicle that will get you where you're going. Once you get there, you won't have any more use for the vehicle, so you leave it behind for someone else. It's best not to choose a total lemon, of course, but other than that the vehicle's 'brand name' doesn't really matter.

Does Buddhism claim to be only way to liberation?

If you're talking about (permanent) liberation from suffering, i.e. Nibbana, then yes, only the Buddha's teaching can result in complete liberation from suffering.

The Buddha teaches in DN 16 (Mahaparinibbana Sutta) that only teachings containing the Noble Eightfold Path can lead to the Four Stages of Enlightenment. In teachings that don't contain the Noble Eightfold Path, Enlightened beings cannot be found.

"Yes, lord," Subhadda answered, and the Blessed One said, "In any doctrine & discipline where the noble eightfold path is not found, no contemplative of the first... second... third... fourth order [stream-winner, once-returner, non-returner, or arahant] is found. But in any doctrine & discipline where the noble eightfold path is found, contemplatives of the first... second... third... fourth order are found. The noble eightfold path is found in this doctrine & discipline, and right here there are contemplatives of the first... second... third... fourth order. Other teachings are empty of knowledgeable contemplatives. And if the monks dwell rightly, this world will not be empty of arahants."

Yes, Buddha Dharma is the only way to attain Nirvana. Nirvana means No Raga, No Dwesha, No Moha permanently, thus no rebirth, no life forms ever again. You cannot find the way to attain Nirvana in any of the other religions from the entire universe.

No, this isn't boasting. This isn't overestimating. This isn't coming from over-believing or stuff along that line.

It is the truth. Because Tripitaka Sutra contains that the Lord Buddha himself said that Arya Ashtangika Margaya — Noble Eightfold Path — is the one and only path to attain Nirvana.

Also the Lord Buddha isn't on the level of other Religious Leaders. The person with a goal to become a Samma Sambuddha has to perform various actions not just 1 day, or 2 days, or a month, or a year, but countless lifetimes. That's one of the reasons why Buddha Dharma is the only way to Nirvana.

And you're right. Buddhism is not for everybody. It's for the fortunate few who has the potential to reach Nirvana. Out of that fortunate few, not everybody attains even Sotapanna phase before they again lose that "Potential" for a long time.

BUT, that doesn't mean Buddha Dharma does not apply to everybody. For example, Karma applies for everybody. Everybody chasing the happiness in this world will taste a ton of suffering and Dhukka sooner or later.

The one with the Mental Illness in your example, became like that because he did something wrong in the previous life. He may have lost the potential to attain Nirvana in this life. But if you were to do one of the following, you can initiate the chain of sequences that will bring him a life that will have the potential he lost this time.

1. If you make them participate in the Donations to Monks, it'll help.
2. If you chant Pirith with him, it'll help.
3. If you apply other Buddhist Pratikarma, it'll minimize the impact of his illness or even cure it + help attain Nirvana someday.

Just like that, Mental Ill guy can too get the benefits of Buddha Dharma + attain Nirvana if he can spark it with things like above.

Above things also apply to the person with Tinnitus. Buddhist Partikarma can help him minimize/cure it. Also Meditation is not required unless the person has attained Sotapanna phase. This means Tinnitus person still can become Sotapanna if he listens to the right Dharma Sermons.

"Prayer" in other religions cannot make someone attain Nirvana. Pirith & Gatha Chanting in Buddhism can be loosely compared to Prayer.

Buddha Dharma applies to everybody from Humans to Dogs to Dewa to Brahma to Ghosts. So it's not unrighteous. Those who became unfortunate to not even hear a Buddha Dharma Sermon became like that because of trying to chase that happiness by doing wrong things.

Buddha isn't responsible for that. But Buddha explained why that happened to them. This way YOU can avoid it happening to you next. YOU can avoid it happening to others.

And yes again, Buddhism is the sole path to Nirvana not just to stop human suffering, but suffering you experience as all life forms such as Deva, Brahma, Ghosts, Animals etc.

To attain Sotapanna, you only need to hear the right Dharma Sermons from a monk who's already attained one of the 4 phases — Sotapanna, Sakrudagami, Anagami, Arihath. To attain other phases, you can start meditation to reach 4th Dhyana and then brainstorm various Dharma Facts as Karmastana.

I recommend learning Buddha Dharma deeper, and then you'll realize what I said is true.

Hope this helps!