Not all inductive inferences are temporal, so the future "resembling" the past can be moot, a more general idea would be that various parts of nature are "uniform", "resemble" each other. But it is not logical to assume that the future will "resemble" the past, or that the nature is "uniform" in this or that aspect. In many cases such assumptions are blatantly unreasonable and contrary to fact, and much better predictions can be made by assuming the opposite, change and non-uniformity. The trick is to tell which is when, that is the real problem of induction. As outlined by Hume and elaborated by Mill the problem is basically that there is no justification for performing inductions: deductive justifications upend the "inductiveness", and inductive justifications are either circular (if there are finitely many types of induction) or lead to infinite regress (if there are infinitely many). Norton's Material Theory of Induction has a nice review of the current state of the issue.
A famous example discussed by Mill concerned the contrast between inductive validity of "some samples of bismuth melt at 271°C, therefore all of them do" and invalidity of "some samples of wax melt at 91°C, therefore all of them do". Mill was so impressed by it that he wrote in his System of Logic:"Whoever can answer this question knows more of the philosophy of logic than the wisest of the
ancients and has solved the problem of induction". It was Mill, who replaced Hume's "custom" with the oft-quoted "axiom of the uniformity of the course of nature", but the melting example demonstrates how "uniformity of nature" is not a solution, but part of the problem. Because the "uniformity" is itself "non-uniform". Norton explains:
"For example, we might require that enumerative induction can only be carried out on A’s that belong to a uniform totality. But without being able to
mention particular facts about bismuth and wax, how are we to state a
general condition that gives a viable, independent meaning to ‘‘uniform’’? We are reduced to the circularity of making them synonyms for ‘‘properties
for which the enumerative induction schema works’’.
[...] All these efforts fall to the problem already seen, an irresolvable tension between universality and successful functioning... if they are general enough to be universal and still true, the axioms or principles become vague, vacuous, or circular. A principle of uniformity must limit the extent of the uniformity posited. For the world is simply not uniform in all but a few specially selected aspects and those uniformities are generally distinguished as laws of nature.
We can argue in broad generalities that "if the nature is not uniform we are in trouble anyway so we might as well optimistically assume it uniform", and even throw in the anthropic principle for good measure ("in universes with non-uniform natures no intelligent creatures can exist"), but it does not provide what a justification must: when it is supposed to work. So at best it is a vague methodological maxim.
P.S. There are many proposals to solve the problem of induction (abductive, Bayesian, Norton's own material one, etc.), but perhaps the best known is Popper's "dissolution" of it. Induction, what induction? Popper's solution to the problem of induction is that there is no induction. What is attributed to induction, according to him, is really a guess followed by a hypothetic deduction of consequences and their corroboration or falsification. Often the guessing is so instinctive and the deduction so trivial (as in say "all crows are black") that we collapse it into undivided "induction". While Popper may have "solved" the problem to his satisfaction, his critics, Norton included, contend that he merely pushed it elsewhere. Because it is unclear what would "justify" entertaining and testing these hypotheses as opposed to infinite others, other than... induction:
"Popper’s account simply fails to bear close enough resemblance to scientific practice if corroboration does not contain a license for belief, with better corroboration yielding a stronger license (see Salmon 1981). Popper does not give much account of the details of the method. The process of conceiving the new hypothesis is explicitly relegated to psychology and the inclination to
take any philosophical interest in it disavowed as ‘‘psychologism’’ (31–
32). So we have only our confidence in the scientist’s creative powers to
assure us that the new hypothesis does not introduce more problems than it
Norton himself looks for solution by localizing induction schemas to specific material domains, where they are licensed by "material facts". This resolves the universality vs functioning dilemma in favor of functioning, but he himself admits that it still creates a regress of "material facts" with unclear prospects of termination. It looks like the problem of induction will remain with us for the foreseeable future.