I wouldn't say entrenched, because I think this is honestly the first time I've seen the two come up together outside of their shared name. I was surprised, but then again sometimes reality is surprising.

Both have function composition, and expressions which contain free variables in multiple places. At the time, that was just a shorthand for what they were trying to express about slight changes. A bit later, formal analysis was axiomised, and is full of infinite things like Cauchy sequences and general topology. In the 20th century, substitution of a composed function into free variables becomes an object of study of it's own, and found to be able to produce full complexity without anything else being added, being Turing equivalent.

All the infinite and continuous stuff that makes calculus work, at least as it's considered abstractly, doesn't really translate into a discrete system. You can numerically approximate it, and I guess you could even use lambda calculus-like functional language to do that, but I'm not mad that never came up in my math courses, like in your original comment.

If there's nothing more to add to that, I am sorry for wasting your time.