Why not use the same distribution for the prior in Bayesian statistics? I am wondering why introductory books on statistics use a conjugate distribution family for the prior instead of using the same pdf of the one we are trying to infer the parameters?
For example, the binomial distribution has the Beta distribution as the prior. They are very similar except for a normalization constant due to the Beta distribution. Why not use another binomial distribution (which can be also uninformative) as the prior?
A case where I see the conjugate prior as necessary is the case when inferring $\Sigma$ of a multivariate Normal given $\mu$ which certainly needs a distribution with symmetrical paramters (e.g. Wishart/Inverse Wishart distribution).

While I can't speak for textbook authors, I can think of two reasons why they might choose to do this.  Both come from the fact that a conjugate prior will lead to a posterior distribution of a known distribution family.
(1) Simplicity: A well-written textbook should distill the content to the key points, and an example where the posterior does not simplify to a convenient form would add clutter that isn't relevant to discussion.  Perhaps the main point could be: the posterior distribution will be a valid pdf or pmf (oh hey, it's a distribution we recognize!)
(2) Computational convenience: With conjugate distributions, algorithms such as Gibbs sampling will be more efficient, as samples can be drawn directly from target distributions without using approximate methods like slice or rejection sampling.
(bonus reason) If the posterior distribution simplifies to a known distribution, closed-form methods can be used rather than MCMC.
In regard to your beta-binomial example, a beta distribution is continuous and bounded on (0,1), which matches the target distribution of the binomial parameter $p$.  Using a binomial distribution as a prior for $p$ wouldn't make sense, because its support is discrete, whole-number values up to and including some parameter $n$.
This is by no means a complete answer, and others are welcome to chime in!
