When to use Binomial Distribution vs. Poisson Distribution?

Poisson distribution

a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

Binomial distribution

the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p

Emphasis mine

For the Poisson you need a known interval (365 days) and a known failure rate (average failures per day - Note: this can be any number $> 0$). For the Binomial you would need a fixed number of trials (365) and a known failure rate per trial (failure chance on a given day Note: this must be a number $\in [0,1]$).

For the specific question, it is a matter of interpretation and both could be justified here.

The Poisson is more appropriate if it is conceivable that the bike could break on a given day, be repaired and break again (and again etc.). For minor failures this is appropriate.

The Binomial is more appropriate if a failure on a given day takes the bike out for the rest of the day (but not for more than that because it would then reduce the total number of days). That is, a moderate failure.

I know from your earlier question here that this is then combined with a Gamma distributed cost - there is no mention of the time the repair takes. If there were, this would be a fairly typical queuing problem which typically uses Poisson distributions. I must say that it was this that led me towards the Poisson.

It's not incorrect to use a binomial distribution because indeed that is what it would be.

However, it is best modeled as a poison distribution because the calculations are much simpler and the approximation is sufficiently close for large $n$

$$\mathcal{Bin}(n, p) \approx \mathcal {Pois}(np), \mbox{ for }n\to\infty$$