It is possible to create a document with an empty field name. E.g. db.col.insert({"":"value"})  However, the driver fails with such documents. For example, it does not work with the BulkWrite API:

var doc = new BsonDocument("_id", 0);
doc[""] = "value";
var model = new ReplaceOneModel<BsonDocument>(
{{ new BsonDocumentFilterDefinition<BsonDocument>(new BsonDocument("_id", doc["_id"])), doc);}}
collection.BulkWrite(new[] { model });

It fails with the following error:
MongoDB.Bson.BsonSerializationException: Element name '' is not valid'.
   at MongoDB.Bson.IO.BsonWriter.WriteName(String name)
   at MongoDB.Bson.Serialization.Serializers.BsonDocumentSerializer.SerializeValue(BsonSerializationContext context, BsonSerializationArgs args, BsonDocument value)
   at MongoDB.Bson.Serialization.Serializers.BsonValueSerializerBase`1.Serialize(BsonSerializationContext context, BsonSerializationArgs args, TBsonValue value)
   at MongoDB.Bson.Serialization.Serializers.SerializerBase`1.MongoDB.Bson.Serialization.IBsonSerializer.Serialize(BsonSerializationContext context, BsonSerializationArgs args, Object value)
   at MongoDB.Bson.Serialization.IBsonSerializerExtensions.Serialize(IBsonSerializer serializer, BsonSerializationContext context, Object value)
   at MongoDB.Bson.Serialization.Serializers.BsonDocumentWrapperSerializer.SerializeValue(BsonSerializationContext context, BsonSerializationArgs args, BsonDocumentWrapper value)
   at MongoDB.Bson.Serialization.Serializers.BsonValueSerializerBase`1.Serialize(BsonSerializationContext context, BsonSerializationArgs args, TBsonValue value)
   at MongoDB.Bson.Serialization.Serializers.SerializerBase`1.MongoDB.Bson.Serialization.IBsonSerializer.Serialize(BsonSerializationContext context, BsonSerializationArgs args, Object value)
   at MongoDB.Bson.Serialization.IBsonSerializerExtensions.Serialize(IBsonSerializer serializer, BsonSerializationContext context, Object value)
   at MongoDB.Bson.Serialization.Serializers.BsonValueSerializerBase`1.Serialize(BsonSerializationContext context, BsonSerializationArgs args, TBsonValue value)
   at MongoDB.Bson.Serialization.IBsonSerializerExtensions.Serialize[TValue](IBsonSerializer`1 serializer, BsonSerializationContext context, TValue value)
   at MongoDB.Driver.Core.Operations.RetryableUpdateCommandOperation.UpdateRequestSerializer.SerializeUpdate(BsonSerializationContext context, BsonSerializationArgs args, UpdateRequest request)
   at MongoDB.Driver.Core.Operations.RetryableUpdateCommandOperation.UpdateRequestSerializer.SerializeValue(BsonSerializationContext context, BsonSerializationArgs args, UpdateRequest value)
   at MongoDB.Bson.Serialization.Serializers.SealedClassSerializerBase`1.Serialize(BsonSerializationContext context, BsonSerializationArgs args, TValue value)
   at MongoDB.Bson.Serialization.Serializers.SerializerBase`1.MongoDB.Bson.Serialization.IBsonSerializer.Serialize(BsonSerializationContext context, BsonSerializationArgs args, Object value)
   at MongoDB.Bson.Serialization.IBsonSerializerExtensions.Serialize(IBsonSerializer serializer, BsonSerializationContext context, Object value)
   at MongoDB.Driver.Core.WireProtocol.Messages.Encoders.BinaryEncoders.CommandMessageBinaryEncoder.WriteType1Section(BsonBinaryWriter writer, Type1CommandMessageSection section, Int64 messageStartPosition)
   at MongoDB.Driver.Core.WireProtocol.Messages.Encoders.BinaryEncoders.CommandMessageBinaryEncoder.WriteSection(BsonBinaryWriter writer, CommandMessageSection section, Int64 messageStartPosition)
   at MongoDB.Driver.Core.WireProtocol.Messages.Encoders.BinaryEncoders.CommandMessageBinaryEncoder.WriteSections(BsonBinaryWriter writer, IEnumerable`1 sections, Int64 messageStartPosition)
   at MongoDB.Driver.Core.WireProtocol.Messages.Encoders.BinaryEncoders.CommandMessageBinaryEncoder.WriteMessage(CommandMessage message)
   at MongoDB.Driver.Core.WireProtocol.Messages.Encoders.BinaryEncoders.CommandRequestMessageBinaryEncoder.WriteMessage(CommandRequestMessage message)
   at MongoDB.Driver.Core.WireProtocol.Messages.Encoders.BinaryEncoders.CommandRequestMessageBinaryEncoder.MongoDB.Driver.Core.WireProtocol.Messages.Encoders.IMessageEncoder.WriteMessage(MongoDBMessage message)
   at MongoDB.Driver.Core.Connections.BinaryConnection.SendMessagesHelper.EncodeMessages(CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Connections.BinaryConnection.SendMessages(IEnumerable`1 messages, MessageEncoderSettings messageEncoderSettings, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.ConnectionPools.ExclusiveConnectionPool.PooledConnection.SendMessages(IEnumerable`1 messages, MessageEncoderSettings messageEncoderSettings, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.ConnectionPools.ExclusiveConnectionPool.AcquiredConnection.SendMessages(IEnumerable`1 messages, MessageEncoderSettings messageEncoderSettings, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Connections.ConnectionExtensions.SendMessage(IConnection connection, RequestMessage message, MessageEncoderSettings messageEncoderSettings, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.WireProtocol.CommandUsingCommandMessageWireProtocol`1.Execute(IConnection connection, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.WireProtocol.CommandWireProtocol`1.Execute(IConnection connection, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Servers.Server.ServerChannel.ExecuteProtocol[TResult](IWireProtocol`1 protocol, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Servers.Server.ServerChannel.Command[TResult](ICoreSession session, ReadPreference readPreference, DatabaseNamespace databaseNamespace, BsonDocument command, IEnumerable`1 commandPayloads, IElementNameValidator commandValidator, BsonDocument additionalOptions, Action`1 postWriteAction, CommandResponseHandling responseHandling, IBsonSerializer`1 resultSerializer, MessageEncoderSettings messageEncoderSettings, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Operations.RetryableWriteCommandOperationBase.ExecuteAttempt(RetryableWriteContext context, Int32 attempt, Nullable`1 transactionNumber, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Operations.RetryableWriteOperationExecutor.Execute[TResult](IRetryableWriteOperation`1 operation, RetryableWriteContext context, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Operations.BulkUnmixedWriteOperationBase`1.ExecuteBatch(RetryableWriteContext context, Batch batch, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Operations.BulkUnmixedWriteOperationBase`1.ExecuteBatches(RetryableWriteContext context, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Operations.BulkUnmixedWriteOperationBase`1.Execute(RetryableWriteContext context, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Operations.BulkMixedWriteOperation.ExecuteBatch(RetryableWriteContext context, Batch batch, CancellationToken cancellationToken)
   at MongoDB.Driver.Core.Operations.BulkMixedWriteOperation.Execute(IWriteBinding binding, CancellationToken cancellationToken)
   at MongoDB.Driver.OperationExecutor.ExecuteWriteOperation[TResult](IWriteBinding binding, IWriteOperation`1 operation, CancellationToken cancellationToken)
   at MongoDB.Driver.MongoCollectionImpl`1.ExecuteWriteOperation[TResult](IClientSessionHandle session, IWriteOperation`1 operation, CancellationToken cancellationToken)
   at MongoDB.Driver.MongoCollectionImpl`1.BulkWrite(IClientSessionHandle session, IEnumerable`1 requests, BulkWriteOptions options, CancellationToken cancellationToken)
   at MongoDB.Driver.MongoCollectionImpl`1.<>c_DisplayClass23_0.<BulkWrite>b_0(IClientSessionHandle session)
   at MongoDB.Driver.MongoCollectionImpl`1.UsingImplicitSession[TResult](Func`2 func, CancellationToken cancellationToken)
   at MongoDB.Driver.MongoCollectionImpl`1.BulkWrite(IEnumerable`1 requests, BulkWriteOptions options, CancellationToken cancellationToken)
   at UserQuery.RunUserAuthoredQuery() in c:\Users\admilazz\AppData\Local\Temp\LINQPad_qtpgpfoi\query_xvftyy.cs:line 57