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It is not clear how you constructed the data but it is clearly wrong. It is not a polynomial but a symbolic expression. Notice the difference in construction between

sage: x = SR.var('x')
sage: p_symb = x^2 + x - 2
sage: x = polygen(QQ)
sage: p_pol = x^2 + x - 2

Or similarly

sage: p_symb2 = SR('x^2 + x - 2')
sage: p_pol2 = QQ['x']('x^2 + x - 2')

You can check that these are indeed different

sage: parent(p_symb)
Symbolic Ring
sage: parent(p_symb2)
Symbolic Ring
sage: parent(p_pol)
Univariate Polynomial Ring in x over Rational Field
sage: parent(p_pol2)
Univariate Polynomial Ring in x over Rational Field

Then, as you already experienced, the symbolic version has no is_irreducible method

sage: p_symb.is_irreducible()
Traceback (most recent call last):
...
AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'is_irreducible'

whereas

sage: p_pol.is_irreducible()
False
sage: p_pol.factor()
(x - 1) * (x + 2)

It is not clear how you constructed the data but it is clearly wrong. It is not a polynomial but a symbolic expression. Notice the difference in construction between

sage: x = SR.var('x')
sage: p_symb = x^2 + x - 2
sage: x = polygen(QQ)
sage: p_pol = x^2 + x - 2

Or similarly

sage: p_symb2 = SR('x^2 + x - 2')
sage: p_pol2 = QQ['x']('x^2 + x - 2')

You can check that these are indeed different

sage: parent(p_symb)
Symbolic Ring
sage: parent(p_symb2)
Symbolic Ring
sage: parent(p_pol)
Univariate Polynomial Ring in x over Rational Field
sage: parent(p_pol2)
Univariate Polynomial Ring in x over Rational Field

Then, as you already experienced, the symbolic version has no is_irreducible method

sage: p_symb.is_irreducible()
Traceback (most recent call last):
...
AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'is_irreducible'

whereas

sage: p_pol.is_irreducible()
False
sage: p_pol.factor()
(x - 1) * (x + 2)

sage: data = [ ['x^2 + x + 1', 'x^2 - 1'], ['x - 1', 'x', 'x^3 - 1']]
sage: R = QQ['x']
sage: p = R(data[0][0])
sage: p
x^2 + x + 1
sage: p.is_irreducible()
True

It is not clear how you constructed the data but and you might have done it is clearly wrong. It is the wrong way. Your data does not seem to be a polynomial but a symbolic expression. Notice the difference in construction between

sage: x = SR.var('x')
sage: p_symb = x^2 + x - 2
sage: x = polygen(QQ)
sage: p_pol = x^2 + x - 2

Or similarly

sage: p_symb2 = SR('x^2 + x - 2')
sage: p_pol2 = QQ['x']('x^2 + x - 2')

You can check that these are indeed different

sage: parent(p_symb)
Symbolic Ring
sage: parent(p_symb2)
Symbolic Ring
sage: parent(p_pol)
Univariate Polynomial Ring in x over Rational Field
sage: parent(p_pol2)
Univariate Polynomial Ring in x over Rational Field

Then, as you already experienced, the symbolic version has no is_irreducible method

sage: p_symb.is_irreducible()
Traceback (most recent call last):
...
AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'is_irreducible'

whereas

sage: p_pol.is_irreducible()
False
sage: p_pol.factor()
(x - 1) * (x + 2)

Considering your input given as a list of lists of strings, you should do something like

sage: data = [ ['x^2 + x + 1', 'x^2 - 1'], ['x - 1', 'x', 'x^3 - 1']]
sage: R = QQ['x']
sage: p = R(data[0][0])
sage: p
x^2 + x + 1
sage: p.is_irreducible()
True