What's the justification for believing that God interacts with us today
the same way He did thousands of years ago?
In a word: Experience.
Consider the value the following men place on experience:
“This grace of God is a very great, strong, mighty and active thing. It does not lie asleep in the soul. Grace hears, leads, drives, draws, changes, works all in man, and lets itself be distinctly felt and experienced. It is hidden, but its works are evident.”
― Martin Luther
“The gospel is not a doctrine of the tongue, but of life. It cannot be grasped by reason and memory only, but it is fully understood when it possesses the whole soul and penetrates to the inner recesses of the heart.”
― John Calvin, Golden Booklet of the True Christian Life
“And Festus said with a loud voice, Paul, thou art beside thyself.”
And so say all the world, the men who know not God, of all that are of Paul’s religion: of every one who is so a follower of him as he was of Christ. It is true, there is a sort of religion, nay, and it is called Christianity too, which may be practised without any such Imputation, which is generally allowed to be consistent with common sense, —that is, a religion of form, a round of outward duties, performed in a decent, regular manner. You may add orthodoxy thereto, a system of right opinions, yea, and some quantity of heathen morality; and yet not many will pronounce, that “much religion hath made you mad.” But if you aim at the religion of the heart, if you talk of “righteousness, and peace, and joy in the Holy Ghost,” then it will not be long before your sentence is passed, “Thou art beside thyself.” ― John Wesley, Sermon 37 "The Nature of Enthusiasm"
In different ways, they are arguing against a 'dead' orthodoxy that merely consists of intellectually assenting to propositions to be 'believed'; but rather for an experiential faith that makes a profound difference to the life of a believer. As these men all believed in sola scriptura, what constitutes 'valid' experience is of course determined by scripture (i.e. 'how God dealt with the ancients') - this necessarily includes 'how God relates to us today'.
The preceding is the substance of my answer, but for the sake of further clarity: From the sermon quoted earlier, Wesley goes on to say, in warning on the dangers of 'enthusiasts' receiving revelation by...
...what is justly called an extraordinary manner: I mean, by visions or dreams, by strong impressions or sudden impulses on the mind. I do not deny, that God has, of old times, manifested His will in this manner; or, that He can do so now: nay, I believe He does, in some very rare instances. But how frequently do men mistake herein! (Ibid., emphasis added)
Wesley at least, affirms that at least some of the 'supernatural' ways God has related to men in the past do still happen, but are not necessarily the norm, rather:
Perhaps some may ask, “Ought we not then to inquire what is the will of God in all things? And ought not His will to be the rule of our practice?” Unquestionably it ought. But how is a sober Christian to make this inquiry? to know what is the will of God? Not by waiting for supernatural dreams; not by expecting God to reveal it in visions; not by looking for any particular impressions or sudden impulses on his mind: no; but by consulting the oracles of God. “To the law and to the testimony!” This is the general method of knowing what is “the holy and acceptable will of God.” (Ibid.)
So this is the norm (as per the model of the Bereans - cf. Acts 17:11), however just because 'visions...dreams...strong impressions or sudden impulses on the mind' from God are 'very rare', they are not non-existant.
It is actually a reasonably common mistake, upon cursorily reading the Bible, to assume that the way God normally interacts with man is overtly miraculous. Such a view (a seemingly implied premise of your question) focuses on the 'spectacular portions' of scripture ignoring the fact that over the historical period covered by scripture, the miracles, visions and dreams etc. presented are actually few and far between (the periods of the Exodus and the ministry years of Jesus being the exceptions that actually prove the rule). Scripture is not just a 'highlights package' of God's 'greatest hits', but it does actually record (significantly extensive) periods where nothing much is happening on the miracle front - the people of God are chronicled as proceeding through life in a fashion that is either characteristically obedient or disobedient to earlier revelation, but not necessarily 'encountering' God until (or sometimes, not even when) some crisis eventuates. Prophets receiving their prophetic revelation may be viewed as a significant exception to this, but for the vast majority of people contemporaneous with the prophets, they could easily have viewed the prophets and what they said as just a guy mouthing words unless there was some miraculous confirmation or direct fulfillment of foretold events (which often didn't happen for generations after the fact) - indeed at least in Jeremiah's case, that is exactly what happened.
In regard to some of the other issues you raise -
Many societies today are vastly different from the societies of the ancients.
Take the rise of democracy, for instance, or the rise of various economic systems
like capitalism and communism. Take the advent of Science and how
electronic/computer technology has become so pervasive and significant. The
controversies surrounding abortion and contraception absolutely could not have
happened two thousand years ago.
The Bible itself portrays:
- People groups that undergo significant cultural transformations - including, but not limited to, the development of various technologies and operating under different political and economic systems - but God's interactions with those people groups not changing significantly as a consequence of these particular changes, but rather -
- God changing the way he relates to people according to how they relate to Him. In the book of Judges for example, viewed at one level, God changes the way He relates to His people from chapter to chapter - alternately allowing them to be oppressed and then delivered (of course, viewed more wholistically, God is completely consistent - it is the people who He deals with that are inconsistent). The Amorites (cf. Genesis 15:6 & book of Joshua) and Ninevites (book of Jonah) are further salutory examples of this.
- Significant and lasting changes in the way He relates to people according to 'Covenants' he makes with them (particularly Adam, Noah, Abraham, Jacob/Israel, Moses & the children of Israel, David and of course the New Covenant offered through Christ to all peoples of the Earth).
Summary: the Bible itself portrays a variety of ways that 'God behaves towards mankind'. Theologians who affirm sola scriptura necessarily bind themselves to scriptural (especially New Covenantal) models of this interaction - for them to deny that God would 'interact with us today the same way He did thousands of years ago' would be inherently inconsistent.