Is it obligatory to thank the husband's father for his financial help to my husband, his claim that he bought me the car, and his boasting about what he provides?

You are not religiously obligated to thank your father-in-law for every good deed he does for your husband. However, if the good deed benefits the family, then your thanking him for it is a good thing, especially concerning the car he bought for you. The Shari'ah encourages rewarding those who do good deeds, and perhaps your thanking him will encourage further assistance. If he boasts about his good deeds, then he should be advised that this may nullify the reward for his action, and it is preferable that this advice comes from people of virtue.