Work term in First Law of Thermodynamics

For the usual simplifying assumptions where the system boundary has negligible inertia, yes, they are the same. For mechanical work by Newton's third law. $$F_\text{system on surroundings}=-F_\text{surroundings on system}$$

and these two forces act at the same location(s). Therefore

$$W_\text{system on surroundings}=-W_\text{surroundings on system}$$

While there are many different systems you can consider (e.g. see FakeMod's answer), in general when the basic form of the first law is described between Chemistry ($\Delta U=Q-W$) and physics ($\Delta U=Q+W$) the two $W$ variables are indeed just related by a negative sign.

To be more precise one would need to be more careful with energy conservation. However, I cannot be more specific without considering more specific scenarios. But at the end of the day work has a precise definition, so when in doubt go back to that.

Referring to mechanical systems, the answer depends on the considered pair of systems and their interactions. If the work is due to contact forces on pairs of contact points with opposite velocities, then the work done by the system on the external system is equal, up to sign, the the one done by the external system to the system as a trivial consequence of the action-reaction principle. That is the case of some gas contained in a cylinder with piston: the system is the gas and the surronding system is the piston.

There are however cases where the contact points have different velocities. It may happen with friction forces: a building block -- viewed as a single material point for the sake of simplicity -- thrown on a rough table. In this case the total work (till the block stops) done by friction force of the table on the block is negative and equals the initial kinetic energy. Conversely, the work on the table (supposed always at rest in the used reference frame) due to the friction force is zero. Here, the contact point changes at each instant but it has always zero velocity.

In summary, the most general way to state the first principle for a system with thermodynamical energy $U$ is $$\Delta U = \cal L+ \cal Q\:,$$ where $\cal L$ is the work on the system and $\cal Q$ is the heat entering the system. This is also valid for irreversible transformations. This is because the principle, in the ideal mechanical case of a purely mechanical system with conservative internal forces, must reduce to a standard theorem of mechanics: the work done on a mechanical system by external forces equals the variation of kinetical energy plus the variation of internal potential energy.