No, like many others you have been confused by the incapacity of those who vote the modifications of the International system of units to decide what kind of units are the units for plane angle and for solid angle: base units or derived units.
A base measurement unit is a unit that is chosen arbitrarily.
A derived measurement unit is one that is determined from the base units by using some relationship between the physical quantity that is measured and the physical quantities for which base units have been chosen.
While there are constraints for the possible choices, the division of the units into base units and derived units is a matter of convention.
Whenever there are relationships between physical quantities where so-called universal constants appear, you can decide that the universal constant must be equal to one and that it shall be no longer written, in which case some base unit becomes a derived unit by using that relationship.
The reverse is also possible, by adding a constant to a relationship, you can then modify its value from 1 to an arbitrary value, which will cause a derived unit to become a base unit for which you can choose whatever unit you like, e.g. a foot or a gallon, adjusting correspondingly the constant from the relationship.
There are 3 mathematical quantities that appear frequently in physics, logarithms, plane angles and solid angles (corresponding to the 1-dimensional space, 2-dimensional space and 3-dimensional space). All 3 enter in a large number of relationships between physical quantities, exactly like any physical quantity.
For each of these 3 quantities it is possible to choose a completely arbitrary measurement unit. Like for any other quantities, the value of a logarithm, plane angle or solid angle will be a multiple of the chosen base unit.
For logarithms, the 3 main choices for a measurement unit are the Neper (corresponding to the hyperbolic a.k.a. natural logarithms), the octave (corresponding to the binary logarithms) and the decade (corrsponding to decimal logarithms).
Like for any physical quantities, converting between logarithms expressed in different measurement units, e.g. between natural logarithms and binary logarithms is done by a multiplication or division with the ratio between their measurement units.
The same happens for the plane angle and the solid angle, for which arbitrary base units can be chosen.
What has confused the physicists is that while for physical quantities like the length, choosing a base unit was done by choosing a physical object, e.g. a platinum ruler, and declaring its length as the unit, for the 3 mathematical quantities the choice of a unit is made by a convention unrelated to a physical artifact.
Nevertheless, the choices of base units for these 3 quantities have the same consequences as the choices of any other base quantities for the values of any other quantities.
Whenever you change the value of a measurement unit you obtain a new system of units and all the values of the quantities expressed in the old system of units must be converted to be correct in the new system of units.
The fact that the plane angle is not usually written in the dimensional equations of the physical quantities in the International System of Units, because of the wrong claim that it is an "adimensional" quantity, is extremely unfortunate.
(To say that the plane angle is adimensional because it is a ratio between arc length and radius length is a serious logical error. You can equally well define the plane angle to be the ratio between the arc length and the length of the arc corresponding to a right angle, which results in a different plane angle unit. In reality the value of a plane angle expressed in radians is the ratio between the measured angle and the unit angle. The radian unit angle is defined as an angle where the corresponding arc length equals the radius length. In general, the values of any physical quantity are adimensional, because they are the ratio between 2 quantities of the same kind, the measured quantity and its unit of measurement. The physical quantities themselves and their units are dimensional.)
In reality, the correct dimensional equations for a very large number of physical quantities, much larger than expected at the first glance, contain the plane angle. If the unit for the plane angle is changed, then a lot of kinds of physical quantity values must be converted.
To add to the confusion, in practice several base units of the 3 mathematical quantities are used simultaneously, so the International System of Units as actually used is not coherent. E.g. the frequency and the angular velocity are measured in both Hertz and radian per second, the rate of an exponential decay can be expressed using the decay constant (corresponding to Nepers) or by the half-life (corresponding to octaves), and so on.